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ANALYSIS OF THE DYNAMIC RESPONSE OF 
TOWERS TO WIND LOADS USING A 
SERIES APPROACH 



By 

GEORGE CHIEN KAO 



A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF 

THE UNIVERSITY OF FLORIDA 
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE 
DEGREE OF DOCTOR OF PHILOSOPHY 



UNIVERSITY OF FLORIDA 

April, 1965 



TO 
Tze-Wei 



ACKNOWLEDGMENTS 

The author wishes to express his sincere appreciation 
to Dr. D. A. Sawyer and Dr. L. E. Grinter, Chairman and Co- 
Chairman respectively of his supervisory committee, for 
their consistent encouragement and guidance throughout the 
entire period of this study. 

He also thanks Professor R. W. Kluge, Head, Depart- 
ment of Civil Engineering, Dr. J. H. Schmertmann, Department 
of Civil Engineering, Dr. C. B. Smith, Department of Mathe- 
matics and Dr. S. Y. Lu, Department of Engineering Mechanics 
for serving on his committee. 

He is grateful to the Department of Civil Engineering 
and the Graduate School for providing a Graduate Assistant- 
snip and a Graduate Fellowship, respectively, during the 
period of his study. 

He wishes to acknowledge the Graduate School for 
making funds available for use in the experiments as 
well as the computer time needed in the research. Thanks 
are also extended to the staff of the University of Florida 
Computing Center for their technical assistance in the pro- 
gramming of the numerical solution. 



iii 



TABLE OF CONTENTS 

PAGE 

ACKNOWLEDGMENTS iii 

LIST OF TABLES „ „ viii 

LIST OF FIGURES x 

KEY TO NOTATIONS ........ ...... xiii 

ABSTRACT xvii 

CHAPTER 

I. INTRODUCTION 1 

1.1. Review of Structural Analysis for 

Wind Loads 1 

1.2. Purpose of the Research .......... 5 

1.3. Limits of the Research . 6 

II. THE FORMULATION OF A MATHEMATICAL MODEL 9 

2.1. Introduction 9 

2.2. Static Characteristics of the Real 
Structure 10 

2.3. Dynamic Characteristics of the Real 
Structure 12 

a) Degree of Freedom of a Truss 12 

b) The Vibration of the Individual Members . 14 

c) Other Effects That May Influence the 
Results 17 

iv 



CHAPTER PAGE 

2.4. The Adopted Mathematical Model 18 

2.5. Methods for Obtaining the Natural 
Modes and Frequencies of the 
Mathematical Model 21 

III. REVIEW OF THE AVAILABLE NUMERICAL PROCE- 



DURES FOR COMPUTING THE DYNAMIC RESPONSE 

OF CONCENTRATED- MASS SYSTEMS . 31 

3.1. Introduction 31 

3.2. Successive Approximation Methods .... 32 



3.3. Series- Expansion Method 34 

3.4. Finite Difference Method , . 35 

3.5. Runge-Kutta Method 36 

3.6. Acceleration Methods 37 

3.6.1. Constant Acceleration Method . „ 39 

3.6.2. Linear Acceleration Method ... 41 

3.6.3. Parabolic Acceleration Method . . 42 

3.6.4. The [3- Method 43 

3.7. General Remarks on the Numerical 
Integration Methods 45 

3.7.1. Errors ............. 45 

3.7.2. Stability 46 

3.7.3. Choice of Numerical Methods ... 47 

3.7.4. Final Comment 47 



v 



J 



J 



CHAPTER PAGE 

IV. THEORY OF THE DYNAMIC RESPONSE OF CON- 
CENTRATED-MASS SYSTEMS , . , 49 

4.1. Free Vibration of a Concentrated- 
Mass System . 49 

4.2. Dynamic Response of a Concentrated- 
Mass System Subject to Dynamic 

Loading 53 

V. THE ADOPTED NUMERICAL SOLUTION FOR THE DYNAMIC 

RESPONSE OF STEEL TOWERS 60 

5.1. Assumptions on the Forcing Function 

Oj^ (t)*oooeeeaoooeeoeee« 61 

5.2. The Derivation of the Numerical Procedure 63 

5.3. The Computer Program .......... 68 

5.3.1. The Variables Defined in the 
FORTRAN Program 69 

5.3.2. The Subroutines Used in the Program 70 

5.3.3. Flow Chart ............ 71 

5.4. Application of the Adopted Method to the 
Design of Steel Towers 75 

VI. EXPERIMENTAL DETERMINATION OF THE NATURAL 

FREQUENCIES OF A K- TRUSSED STEEL TOWER 76 

6.1. Purpose of the Experiment 76 

6.2. The Test Model . 76 

6.3. Instruments Used in the Experiment ... 77 

6.4. The Testing Procedure 82 

6.5. The Theoretical Flexibility Factors 
and the Concentrated Masses of the Test 
Tower ... 83 



vi 



CHAPTER PAGE 

6.6. The Experimental Determination of the 
Flexibility Factors .......... 86 

6.7. The Rotation of the Support ...... 87 

6.8. Test Results 91 

6.9. Discussions of the Test Results .... 98 

VII. NUMERICAL ANALYSIS OF STEEL TOWERS SUBJECT 

TO WIND 102 

7.1. The Three-Mass System 103 

7.2. The Six-Mass System ......... .111 

7.3. The Eighteen-Mass System 117 

VIII. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS FOR 

FUTURE RESEARCH 174 

8.1. Summary 174 

8.2. Conclusions ...... , 176 

8.3. Recommendations for Future Research . 180 

BIBLIOGRAPHY 184 

APPENDIX A , 188 

APPENDIX B ..... 206 



BIOGRAPHICAL SKETCH . 217 



vii 



LIST OF TABLES 



J0^ 



TABLE PAGE 

1. DEAD WEIGHTS AND MASSES AT EACH PANEL 

POINT . . 85 

2. ROTATIONAL STIFFNESS OF SUPPORT ...... 88 

3. DISTRIBUTION OF MASSES (TEST NO. 1) ... , 93 

4. ANALYTICAL AND EXPERIMENTAL NATURAI FRE- 
QUENCIES IN RAD. /SEC. (TEST NO . 1 ) .... 94 

5. DISTRIBUTION OF MASSES (TEST NO . 2 ) . . . . 95 

6. ANALYTICAL AND EXPERIMENTAL NATURAL FRE- 
QUENCIES IN RAD. /SEC. (TEST NO . 2 ) .... 96 

7. DISTRIBUTION OF MASSES (TEST NO. 3) ... . 97 

8. ANALYTICAL AND EXPERIMENTAL NATURAL FRE- 
QUENCIES IN RAD. /SEC. (TEST NO . 3 ) .... 98 

9. WIND VELOCITIES AT EACH MASS POINT 
(THREE-MASS SYSTEM) 106 

10. WIND VELOCITIES AT EACH MASS POINT 
(SIX-MASS SYSTEM) 113 

11. WIND VELOCITIES AT EACH MASS POINT 

( EIGHTEEN-MASS SYSTEM) 125 

12. THE DISPLACEMENT RESPONSES OF MASS 1 ... 128 

13. THE DISPLACEMENT RESPONSES OF MASS 2 ... 130 

14. THE DISPLACEMENT RESPONSES OF MASS 3 ... 132 

15. THE DISPLACEMENT RESPONSES OF MASS 4 ... 134 

16. THE DISPLACEMENT RESPONSES OF MASS 5 ... 136 

17. THE DISPLACEMENT RESPONSES OF MASS 6 ... 138 

18. THE DISPLACEMENT RESPONSES OF MASS 7 ... 140 

viii 



TABLE PAGE 

19. THE DISPLACEMENT RESPONSES OF MASS 8 142 

20. THE DISPLACEMENT RESPONSES OF MASS 9 144 

21. THE DISPLACEMENT RESPONSES OF MASS 10 .... . 146 

w 

22. THE DISPLACEMENT RESPONSES OF MASS 11 148 

23. THE DISPLACEMENT RESPONSES OF MASS 12 150 

24. THE DISPLACEMENT RESPONSES OF MASS 13 152 

25. THE DISPLACEMENT RESPONSES OF MASS 14 154 

26. THE DISPLACEMENT RESPONSES OF MASS 15 .... . 156 

27. THE DISPLACEMENT RESPONSES OF MASS 16 .... . 158 

28. THE DISPLACEMENT RESPONSES OF MASS 17 160 

29. THE DISPLACEMENT RESPONSES OF MASS 18 .... . 162 

J 



ix 



LIST OF FIGURES 

FIGURE PAGE 

1. MATHEMATICAL MODEL OF A K- TRUSS ED 

TOWER AND ITS CONCENTRATED FORCES ..... 20 

2. THE MATHEMATICAL MODEL OF A STEEL TOWER , , 22 

3. EULER'S SUCCESSIVE APPROXIMATION METHOD . . 33 

4. CONSTANT ACCELERATION METHOD o 39 

5. LINEAR ACCELERATION METHOD . 41 

6. PARABOLIC ACCELERATION METHOD 43 

7. CONSISTENT VARIATIONS OF ACCELERATION IN 



A TIME INTERVAL ......... 45 

8. CONCENTRATED-MASS SYSTEM SUBJECTED TO 

DYNAMIC FORCES ... ...... .53 

9. A TWO-MASS SYSTEM . 54 

10. LAGRANGIAN INTERPOLATION POLYNOMIAL .... 67 

11. DETAIL CONSTRUCTION OF THE TEST TOWER ... 78 

12. PRINCIPLE OF THE OSCILLATOR „ 79 

13. EXPERIMENTAL ARRANGEMENT 80 

14. SCHEMATIC DIAGRAM OF THE TEST CIRCUIT ... 81 



15. THE IDEALIZED SYSTEM OF THE TEST TOWER. . , 83 

16. FASTENING OF LEAD WEIGHT ON THE TEST TOWER. 92 

17. FASTENING OF RAILS ON THE TEST TOWER ... 92 



x 



LIST OF FIGURES (CONTINUED ) 



FIGURE PAGE 

18. INFLUENCE OF SUPPORT FLEXIBILITY 
ON THE NATURAL FREQUENCIES OF 

MATHEMATICAL MODEL X01 

19. THREE-MASS SYSTEM . 103 



20. DISPLACEMENT RESPONSES OF THE THREE- 
MASS SYSTEM 



107 



21. EFFECT OF COMBINATION OF MODES ON THE 
DISPLACEMENT RESPONSES OF MASS 1. . . . 108 

22. EFFECT OF COMBINATION OF MODES ON THE 
DISPLACEMENT RESPONSES' OF MASS 2 . . . 109 

23. EFFECT OF COMBINATION OF MODES ON THE 
DISPLACEMENT RESPONSES OF MASS 3 ... X10 

24. SIX-MASS SYSTEM ............ m 

25. DISPLACEMENT RESPONSES OF THE SIX-MASS 
SYSTEM 115 

26. EFFECT OF COMBINATION OF MODES ON THE 
DISPLACEMENT' RESPONSES OF MASS 6 ... 216 

27. THE EIGHTEEN- MASS SYSTEM ....... 217 

28. DISPLACEMENT RESPONSES OF THE EIGHTEEN- 
MASS SYSTEM 164 

29. EFFECT OF COMBINATION OF MODES ON THE 
DISPLACEMENT RESPONSES OF MASS 1 ... 265 

30. EFFECT OF COMBINATION OF MODES ON THE 
DISPLACEMENT RESPONSES OF MASS 2 ... 266 

31. EFFECT OF COMBINATION OF MODES ON THE 
DISPLACEMENT RESPONSES OF MASS 3 ... 167 

32. EFFECT OF COMBINATION OF MODES ON THE 
DISPLACEMENT RESPONSES OF MASS 4 . . . i^q 



xi 



LIST OF FIGURES (CONTINUED) 



FIGURE PAGE 

33. EFFECT OF COMBINATION OF MODES ON 

THE DISPLACEMENT RESPONSES OF MASS 9 . . 169 

34. EFFECT OF COMBINATION OF MODES ON THE 
DISPLACEMENT RESPONSES OF MASS 15 ... . 370 

35. EFFECT OF COMBINATION OF MODES ON THE 
DISPLACEMENT RESPONSES OF MASS 16 ... . 171 

36. EFFECT OF COMBINATION OF MODES ON THE 
DISPLACEMENT RESPONSES OF MASS 17 ... . 172 

37. EFFECT OF COMBINATION OF MODES ON THE 
DISPLACEMENT RESPONSES OF MASS 18 ... . 173 



xii 



KEY TO NOTATIONS 



a i j Coefficient in a linear simultaneous frequency 

equation 

A Cross-sectional area of steel member in a truss 

Aj The lumped projected wind area at mass j, or the 

cross- sectional area of truss member K whenever 

defined 

[ B] A rectangular matrix defined by Eq. (2.11) 

[ B]" 1 Inverse of [b] 

[BC] Product of [B] and [ C j 

[C ] A rectangular matrix defined by Eq. (2.11) 
C d Drag coefficient of the projected wind area 

assigned to the corresponding mass point 
d Flexibility coefficient 

d! ' Experimental flexibility coefficient 

A dii Difference between d. and d. 
J ij ij 

d. Modified flexibility coefficient 

( A Correction of d^j due to support rotation 

A square matrix; each element contains the 

product of m . d . . 

i ij 

D m (t) Dynamic factor of m th mode due to external dynamic 
force 



xiii 



KEY TO NOTATIONS (CONTINUED) 

D m (t) Dynamic factor of rrT n mode due to initial motion 
E Modulus of elasticity (29,000,000 psi) 

f(x,y) A function of variables x and y, where x is 

an independent variable and y a dependent 

variable 

F(t) External dynamic force at time t 
F Alternating force generated by an oscillator 

fJ Matrix of static equivalent force defined by 

Eq. (5.14) 
h Increment in x or time t 

Li] Unit matrix defined by Eq. (2.13) 
Ij(t) An integrand defined by Eq. (5.6) 

Stiffness matrix of a mathematical model 
kj_ Wind conversion factor at mass point i 

k 1 » ^2 , k^ , k^ 

Constants defined in Runge-Kutta method 
k Rotational stiffness of support 

^•jm Characteristic load at mass point j for the 

m^ mode 

L Length of a steel member in a truss 

m Mass 

rru Lumped mass at panel point i 

N Speed of an oscillator in rpm 

P ir A function of order frequency equation 



xiv 



KEY TO NOTATIONS ( CONTINUED ) 



P m Participation factor for the n th mode 

Pj(t) Dynamic wind pressure at mass point j 

Maximum magnitude of Q^(t) 
Ql Equivalent static wind load acting at mass 

point i 

Qjjt) Equivalent dynamic wind load acting at mass 

point i at time t 

r. x, /x, 

lm lm 

R n Remainder term in series expansion method 

R(x) Restoring force which is a function of 

displacement x 
S Stress in steel member of a truss 

1.5 'J A sweeping matrix defined by Eq. (2.13) 

Force in truss member i due to a unit load 
applied at panel point j 
V\ Concentrated wind velocity at mass point i 

| xlj Displacement matrix of the extreme position 

Xjl Displacement of mass i 

Xj_ Velocity of mass i 

x Acceleration of mass i 

x i(j) j th trial natural ordinate at mass point i 
X im Natural ordinate of mass point i at m th mode 

X im(j) x im^)A 1m (j) 



im VJ/ lm v J 'f A i m 
z °(t) Displacement response of mass i due to initial 
motion at time t 

xv 



KEY TO NOTATIONS (CONTINUED) 

x^(t) Displacement response due to external dynamic 
force at mass i at time t 

Yi ^ The j th trial value of the product of the 

matrices D and x , or j th trial value of a 
linear differential equation 

Y i (j) Ration of y i (j)/y i (j) 

y( n ) n tn derivative of y 

(y i )( j ) j tn trial value of y- 

7 y Backward dif f erences 

A rectangular null matrix defined by Eq. (2.10) 

A parameter used in Newmark ' s method 
(d^) support/( d-^ )truss 

Angle between eccentric weights of an oscillator 
j th eiger root of an r order equation 
(<° n ) flexible base/(^ n ) fixed base 
Normal ordinate of mass point i at m t '" 1 mode 
Natural frequency of steel tower 



T 
9 



f 



? 1 



im 



x vi 



Abstract of Dissertation Presented to the Graduate Council 
in Partial Fulfillment of the Requirements for the 
Degree of Doctor of Philosophy 

ANALYSIS OF THE DYNAMIC RESPONSE OF TOWERS TO 
WIND LOADS USING A SERIES APPROACH 

By 

George Chien Kao 

April, 1965 

Chairman: Dr. D. A. Sawyer 
Co-Chairman: Dr. L. E. Grinter 
Major Department: Civil Engineering 

A numerical solution for computing the displacement 
responses of lattice-type steel towers, cantilevered from 
their bases and free from any intermediate supports was 
studied. The real structures were first idealized into 
mathematical models which possessed the same linear elastic 
properties as the original structures but with all masses 
concentrated (lumped) at panel points. The wind pressures 
expressed either analytically or numerically were assumed 
to act discretely at the panel points. Damping was not 
considered in the analysis. The numerical solution 
presented utilizes the normal modes and their associated 
natural frequencies of the mathematical model as well as 
the derivatives of the forcing functions. Three numerical 
examples are illustrated. Their results indicate that the 
use of the lowest three modes provide accurate approxima- 
tions for the displacement responses. 



xvii 



Experiments were conducted on a test tower 
fifteen feet long (K-truss type) with a rectangular cross 
section of 2 ft.-O in. by 1 ft. -4 in. The tests showed 
that if non- structural masses were rigidly connected 
to the structural members at panel points the first two 
natural frequencies of the test tower could be determined 
adequately from its mathematical model. The error was 
1 per cent in the fundamental mode and 3 per cent in the 
second mode. Higher modes were not verified because of 
the limitations of the test equipment. 



xviii 



CHAPTER I 
INTRODUCTION 

1.1. Reyj.ew of Structural Analysis for Wind Loads 

The current design procedure for steel towers 
subject to wind load is based on a set of equivalent 
static wind loads, Q^/ acting at each panel point i 
of the tower. Q i can be expressed by Eq. (l.l) 

Qi* ■ Ki Vi 2 

where Vj_ is the equivalent wind velocity at panel 
point in miles per hour and Ki is a constant de- 
termined by the shapes, the shielding effects, the 
lift and drag coefficients of the structural members. 
The stresses which result from the static equivalent 
loads are used to select member sizes according to 
specifications. 



*The symbols adopted for use in this dis- 
sertation are defined where they first appear and 
are listed alphabetically in Key to Notations. 



2 

The velocity used in the design is 
determined by an equation which defines a velocity 
profile based on local geographical and climatol ogical 
conditions. Recommendations for establishing a suit- 
able wind profile to determine the maximum design 
velocity have been suggested by Sherlock (28), 
Collins (8). Saffir (25). Thorn (35), Van Erp (37). 
and Scruton (27), but none of these scholars ever 
considered the dynamic effect of wind pressure on 
the structure itself except by the simplified idea 
of including a gust factor in the design wind velocity. 
Data obtained by Sherlock (28) and the Brooklyn 
National Laboratory (29 to 34) indicated that the 
velocity of the wind varies not only with the height 
but also with the time in a random manner. Due to 
the fluctuating nature of the wind, the dynamic stresses 
are sometimes greater than those determined by the 
conventional static equivalent load. 

The report '(2) presented by the Task Committee 
on Wind Forces of the American Society of Civil Engi- 
neers strongly emphasized the importance of dynamic 

- 

Numbers within parentheses throughout this 
dissertation refer to the Bibliography. 



effects of wind on tall structures. However, no 
method for solving such problems was mentioned. 
With the ever increasing tendency toward the use 
of tall steel towers to meet today's complex in- 
dustrial needs, a design method that takes the 
dynamic characteristics of wind load into consider- 
ation is essential to provide a reliable structure. 

The dynamic response of structures to 
wind load has been investigated by Davenport (10, 11) 
and Chiu (7). Davenport used a stationary time 
series to formulate the wind load. The responses 
were expressed in terms of the mean wind velocity, 
the velocity spectrum or the gustiness, and the 
mechanical and aerodynamic properties of the structure. 
But his solution could apply only to a continuous 
system or a system with only one degree of freedom. 
Chiu's method employed a mathematical model of one 
degree of freedom with a natural fundamental period 
and a spring constant equivalent to that of the actual 
tower. A statistical power- law distribution for the 
vertical wind profile was used in the analysis of 
the dynamic responses. Because higher modes and fre- 
quencies were neglected in Chiu's method, one could 
not get a complete picture of the deflected tower, and 



doubts were raised about the accuracy of the 
simplified system. 

To obtain the instantaneous dynamic stresses 
of a steel tower due to wind load, the actual dis- 
placements for the adjacent panels are needed. A 
suitable mathematical model to achieve this is pro- 
vided by lumping the mass of the tower at its panel 
points while its elastic properties remain unchanged. 
Thus, the relative displacement of the lumped masses 
will yield the basis for a good approximation of the 
dynamic stresses. The wind pressure acting on each 
lumped mass could be represented by either a mathe- 
matical function or the actual data recorded from a 
storm. The equations of motion of the lumped-mass 
system thus defined are solved mainly by numerical 
methods. The analytical methods are difficult to ap- 
ply because of the complicated nature of the wind 
function. The numerical methods include the successive 
approximation method (26), Taylor series method (24), 
Milne's method (16), finite difference method (3, 5), 
Runge-Kutta 's method (24), and acceleration method 
(24). The 6 - method (22) as developed by Newmark 
is widely used for complicated structures. 



The advantage of using a numerical method 
is that once the equations of motion of a structural 
system are formulated, the solution can be obtained 
in a straightforward manner; the disadvantage is that 
its physical characteristics such as natural fre- 
quencies and normal modes are not considered explicit- 
ly. In other words, the numerical method is purely 
a mathematical manipulation. The accuracy of the 
numerical solution will depend on the time interval 
chosen in the computation as well as the given initial 
conditions. However, all modes of the lumped-mass 
are free to affect the results. 

1.2. Purpose o f the Research 

The main purpose of the research is to de- 
velop a numerical method based on a suitable mathe- 
matical model for estimating the dynamic response of 
steel towers to wind load. The influence of the 
dynamic characteristics, i.e., the normal modes and 
their associated natural frequencies, will be of 
major interest in the investigation. The proposed 
method differs from those described above in two 
respects, namely. 



a) the natural frequencies and the normal modes are 
being considered explicitly in the course of the 
analysis, and 

b) the accuracy of the response will depend on the 
number of natural frequencies and the correspond- 
ing normal modes used. 

It was considered important that the de- 
signer should be able to decide in advance as to how 
many terms would be required in the computation to 
achieve a solution of the desired accuracy in a 
reasonable amount of computational time. An experi- 
ment was made during this study to verify the as- 
sumption that the natural frequencies of an actual 
structure are reasonably close to those of an equiva- 
lent lumped-mass system used in the analysis. 

It is assumed that the solution obtained 
from the proposed method would be helpful for design- 
ing actual structures. The availability of a high- 
speed electronic computer is implied. 

Limits of the Research 

The research is confined to the study of 
the dynamic displacement response of free-standing, 
latticed steel towers due to time- dependent wind 
loads. The mathematical model of the steel tower 



is a vertical cantilever structure, fixed at the 
base with its distributed mass lumped at discrete 
points which correspond to the panel points of the 
real truss-type tower. The idealized structure 
possesses the same elastic properties as the original, 
and the natural frequencies are obtained by applying 
the Stodola-Vianell o procedure. 

The dynamic displacements of the concen- 
trated masses are assumed to be extremely small as 
compared to the height of the actual tower; there- 
fore, the linear elastic theory may be applied, and 
the displacement response due to different loadings 
can be superimposed. 

No torsional response of the actual tower 
was considered. The wind pressure, which is taken 
as a piecewise continuous function of time t, is as- 
sumed to be acting normally at each concentrated 
mass. The wind force at each point is taken to be 
the same force that would act on the corresponding 
panel point of the real structure. It is also assumed 
that the frictional damping as well as the structural 
damping (9, 14, 23) are small: therefore, they are 
not considered in formulating the equation of motion. 



The entire computation has been performed 
on an IBM 709 computer. The programs used in the 
computation are listed in Appendix A. 

In order to validate the application of the 
Stocol a-Vianel 1 o procedure to truss-type structures, 
a small steel tower fifteen feet long was tested for 
its natural frequency. Two sets of concentrated 
masses were also attached for two of the tests. Al- 
though the results obtained were limited to some 
extent by the available vibration-generating equip- 
ment, they were considered quite useful. 



CHAPTER II 



THE FORMULATION OF A MATHEMATICAL MODEL 
2.1. Introductio n 

In order to analyze a complicated structure 
for a specific problem, it has been the practice of 
engineers to formulate a mathematical model to repre- 
sent the true structure under simulated conditions. 
The mathematical model is usually obtained by making 
certain idealizations by which the problem can be 
solved without the influence of other secondary 
factors. Furthermore, the mathematical model should 
be simple enough to be analyzed but it must still re- 
tain the original characteristics of the true struc- 
ture so that its response can be used to predict the 
response of the real structure. 

The purpose of this dissertation is to find 
a solution for the displacement response of latticed 
tower structures; therefore, in formulating the 
mathematical model, only those factors which con- 
tribute primary characteristics to the structure will 
be taken into consideration. But before reaching a 



9 



10 

final decision on selecting a model, it is necessary 
to consider the static and dynamic characteristics 
of the real structure. 

2.2. Static Characteristics of the Real Structure 
A steel tower is generally a space frame 
composed of vertical trusses which are arranged sym- 
metrically with respect to its vertical axis. In 
order to evaluate the static reaction of each member 
in the truss, the following assumptions are made; 

a) The members are connected by frictionless pins. 

b) The cross-section of each member is taken as 
c onstant . 

c) The elongation of each bar is given by SL/AE, 
where S « force in the member 

L = length of the member 
A = area of the member 
E ■ modulus of elasticity of steel 
(29,000,000 PSi) 

d) The loads are applied only at the joints. 

e) The lenght of each member is taken as the distance 
between the centers of its end joints. 

f) The response is linear. 



Under these assumptions, the members of the tower 
are ideal "two- force" members and are subject only 
to tensile or compressive forces. However, the 
construction of a real truss deviates from the 
original assumptions in the following ways: 

a) The joints are formed by means of gusset plates 
to which structural members are either welded, 
riveted, or bolted. Therefore, the joints are 
more nearly rigid than f rictionless . 

b) The chord members are frequently continuous for 
several panel lenghts. The continuity of these 
members increases the rigidity of the joints. 

c) The lines through the centers of gravity of in- 
dividual members may not intersect at a single 
point. The effect of the eccentricity introduces 
secondary moments into all the members converg- 
ing at that joint. 

The displacements of a tower under static loads induce 
bending stresses as a result of (a) and (b) stated 
immediately above. These bending stresses together 
with those induced by (c) constitute the secondary 
stresses in all' the members. In general practice, 
the magnitude of secondary stress in each member 
should not exceed 30 per cent of the direct stress 



12 

otherwise the section of the member must be rede- 
signed. In addition to the bending stresses pro- 
duced by (a), (b), and (c), the attachment of non- 
structural weights such as equipment or machines 
should also be investigated, so that the operating 
conditions will not deviate too far from those as- 
sumed. 

2.3. Dynamic Characteristics of the Real Structure 
a) Degrees of Freedom of a Truss 

In considering the vibrations of trusses, 
there are two types of oscillations that could occur, 
namely: vibration of the truss as a whole in which 
all members act as ideal or straight two- force members 
during the motion; and, vibrations of the individual 
members between their end joints. This study is primarily 
concerned with the response of a truss as a complete 
structure so that the first type of oscillation is of 
major interest in this study. The characteristics of 
individual member vibrations and their effects on the 
primary type of vibration will be discussed in the 
following section. 



During truss-type vibrations it is assumed 
that all members act as two-force members so that the 
distortion of the truss is a function only of the ex- 
tensions and contractions of its component bars. For 
a space truss, the original and distorted shapes can 
be completely defined by the x- , y- , and z-c oordinates 
of the joints. Therefore, it is apparent that the 
maximum number of degrees of freedom of a space truss 
can be taken as three times the number of its joints. 
For uniform towers that possess appropriate configura- 
tions, it is possible to treat the truss system as a 
combination of plane trusses so that only two coordi- 
nates will be required at each joint to completely 
define the displacements. Hence, for plane trusses, or 
space trusses that can be treated as plane trusses, the 
number of degrees of freedom may be taken as two times 
the number of joints. Further, for vertical towers, the 
horizontal movements of the joints are several times 
as great as the vertical movements. Also most 
tower structures vibrate on their lowest modes under 
wind pressures. Therefore, the inertia effects caused 
by the vertical joint movements are doubtlessly very 
small compared to the inertia effects caused by the 
horizontal displacements. Because of these factors. 



14 

the position of each joint can be satisfactorily des- 
cribed by a single coordinate. Finally, for vertical 
towers in which all joints in each panel are at the 
same elevation, the difference in the horizontal move- 
ments of each joint of a panel is equal to the extension 
of the connecting horizontal member. This difference is 
quite small compared to the total movements of the corre- 
sponding joints, particularly for the lower modes of 
vibration. Therefore, as a final simplification, it is 
assumed that the number of degrees of freedom of a tower 
may be taken as the number of panels in the component 
trusses. This assumption is not expected to cause great 
error for the lowest few modes of vibration but it may 
cause significant errors as the mode number approaches 
the number of panels in the trusses. 

*>) Ihe Vibration of Individ ual Mgmhgrs 

As noted In the section above, the individual 
members of a truss possess their own vibration charac- 
teristics. In the completely general sense, the truss- 
type vibration discussed above and the individual member 
vibrations are not uncoupled. Theoretically, the vi- 
bration of the truss as a whole could affect its compo- 
nent members in the following ways: 

1.) The periodic vibration of the truss induces peri- 
odic axial forces in the individual members that 



could cause stress magnifications due to resonance 
of longitudinal vibrations if the frequency of the 
truss vibration and the natural frequency of 
longitudinal vibration of any member happened to be 
sufficiently matched. 

The periodic axial force induced in a member by 
the truss vibration could cause large transverse 
flexural vibrations to occur in the same member 
if the frequency of the axial force happens to be 
close to two times the natural frequency of trans- 
verse vibration of the member considered. This 
has been treated in some detail by Bolotin (4). 
The axial forces induced in the truss members 
will change the frequencies of transverse vibration 
in these members. Axial compression will lower the 
natural flexural frequency whereas a tensile force 
will increase the natural flexural frequency. If 
the axial force changes sense during the vibration 
of the truss, the effects are highly non-linear 
and are extremely difficult to evaluate. 
The transverse vibrations of the truss will cause 
transverse inertial forces on the individual mem- 
bers. These transverse forces can induce resonant 
transverse vibrations in the individual members if 
the frequency of truss vibration closely matches a 
natural mode of any member considered. 



Further, the natural frequencies of vibration (longi- 
tudinal or transverse) of the individual members are 
influenced greatly by their end conditions. Actual truss 
connection details cause the end conditions to be some- 
what indefinite. 

Because the elastic properties of a truss depend 
on the deformation characteristics of its individual mem- 
bers, any of the effects listed above could influence 
the stiffness and response of a truss by causing the 
individual members to deform differently from SI/AE as 
assumed in the computation of the flexibility matrix. 
Therefore, in the worst case there could be considerable 
interaction between the vibration of the truss and the 
vibration of its individual members. This interaction 
could render the analysis highly complex or, indeed, 
entirely impractical. 

In the present study we choose to ignore these 
complicating factors because it is possible to design a 
structure to avoid them. In most practical cases, it is 
sufficient to compute the natural frequencies based on 
the flexibility matrix calculated on the assumption of 
ideal static truss action. The natural frequencies of 
vibration of each individual member should then be calcu- 
lated for comparison with the truss frequencies. In most 
cases, it will be found that the individual member 



17 

frequencies are much higher than the frequencies of over- 
all truss vibration so that resonant effects will not 
occur. The studies of truss responses to natural winds 
as given in Chapter VII show that towers will vibrate 
primarily in the first mode so that resonant effects 
under real conditions are almost never a problem, 
c) Other Effects That May Influence the Results 

In addition to the factors discussed above 
there are several other effects that could influence the 
results and are mentioned here for completeness. Because 
ideal truss action requires all loads to be applied only 
at the joints whereas the mass of a truss is distributed 
along its individual members, it is obvious that the 
mathematical model finally adopted must involve "lumping" 
or concentration of the masses. The most obvious points 
at which lumping should be considered are the truss 
joints. More particularly, if one assumes that the 
number of degrees of freedom of a cantilever truss can 
be taken as the number of panels in that truss, one-half 
of the masses of individual members should be lumped at 
the joint in each panel where the x-c oordinates are to 
be computed during the dynamic response calculations. 
Obviously, such a simplified lumping will completely dis- 
tort the mass moments due to the inertias of the individual 
members. By analogy with beam-type problems, the effect 



considered here may be called the "rotary inertia" effect. 
The magnitude of the error produced will vary according 
to the center about which each truss member tends to 
rotate during the vibration of the truss. For the lowest 
modes of vibration the center of rotation of the members 
will be located far from their centers of vibration. In 
such cases, the error will be small. For the highest 
modes of vibration, each member will tend to rotate about 
a point that lies somewhere between its ends and the 
error will be relatively large. Therefore, the lumping 
of the mass at a particular joint in each panel will cause 
percentage errors that will increase as the mode number 
increases. Because the studies of Chapter VII show that 
towers subjected to natural winds tend to vibrate in a 
shape that is close to the fundamental mode, the neglect 
of rotary inertia effects seems justifed. 

2.4. The Adopted Mfft ha mtical Mop^ l 

The customary mathematical model that has gained 
considerable recognition in the study of complicated 
structures is a finite- degree- of- freedom system that pos- 
sesses the same elastic properties as the original struc- 
ture but which lumps the mass at certain discrete points. 
The number of lumpings determines the number of degrees 
of freedom of the mathematical model. For continuous 



19 

structures such as beams* one has a free choice as to 
the number of points at which the mass may be lumped. 
It has been established that for such structures 

one must lump the mass at approximately twice as many 
points as the highest mode of vibration for which accept- 
able accuracy is desired. However, for truss-type struc- 
tures one does not have perfect freedom of choice as to 
where the mass may be lumped. To satisfy ideal truss 
action, any point chosen should coincide with one of the 
truss joints (19) (38). In the present case, it was con- 
sidered satisfactory to lump the mass at the joints 
between panels so that the number of degrees of freedom 
of the mathematical model becomes equal to the number of 
panels in the tower trusses. This assumption as well as 
ail others that were made during the formulation of the 
model are as follows: 

a) The mathematical model possesses the same elastic 
properties as the original structure. In other 
words, the elastic properties of the model are 
given by the flexibility matrix of the ideal 
truss, so that all the assumptions made in 
section 2.2 hold. 

b) Relative displacements of joints on the same 
level are neglected and their displacements are 
considered to be identical. The masses, both 



structural and non-structural types, are lumped 
at the center joint of the h-type panel. 

c) Vertical displacements of masses together with 
their vertical inertia effects are neglected. 

d) The number of degrees of freedom of the model is 
equal to the number of panels in the real struc- 
ture. 

e) The dynamic loads act on the lumped masses at 
the panel points. 

f) The rotary inertia effects of individual members 
are neglected. 

■ 

The mathematical model for a four-panel K-truss tower is 
shown in Fig. i. 




m, 



m 2 

m 3 <H 



Fig. 1. MATHEMATICAL MODEL OF A K- TRUSSED TOWER 
AND ITS CONCENTRATED FORCES 



21 

The analytical investigations (12,13,15) on the 
similarity between the lumped-mass model and the real 
structure are limited to beam structures. The dynamic 
characteristics of the adopted model for the tower struc- 
ture will be checked experimentally in Chapter VI, 

2 - 5 - Methods for Qbjajj^ ag the Natural Modes and 
the Frequen cies of the Mathematical Model 

The mathematical methods developed during 
recent years for calculating the natural frequencies of 
lumped mass systems includes the Holzer-Myklestad 
method (18), the Stodola-Vianello method (24), and the 
escalator method (17). The Holzer-Myklestad method is 
primarily designed for close-coupled systems (Holzer) or 
continuous flexural systems (Myklestad). A modified 
version applying to truss systems is being studied under 
another investigation, and will not be considered here. 
We will discuss briefly the formulation of the Stodola- 
Vianello method and the escalator method in the following 
paragraphs. 

The Stodola-Viane llo M ethod 

The Stodola-Vianello method is an iterative method 
which utilizes the flexibility matrix or the stiffness 
matrix of the mathematical model. The approach using the 
flexibility matrix is given here. 



22 

The mathematical model for a steel tower is 
idealized into a concentrated-mass system consisting of 
a cantilevered elastic structure with finite degrees of 
freedom as shown in Fig. 2. 




FIG. 2. THE MATHEMATICAL MODEL OF A STEEL TOWER 

Each represents the lumped-mass at the panel point i, 
and the cantilevered elastic structure is assumed to be 
weightless but to possess the same linear-elastic prop- 
erties as the original structure. That is, the flexi- 
bility matrix of the cantilevered elastic structure is 
taken to be the same as the flexibility matrix of the 
original steel tower. 

For free vibration, the displacement at each mass 
point can be expressed in a matrix form as 



y If) \ 

x 2 (t) 




m A m m 

• 

0 

o 

0 


m di ^ 

9 


< 


' xi (t)l 
x 2 (t) 

; 


X (t) 

n 




• 

m d ■■ 
n nl 


m„d„„ 
n nn 




X (t) 

n 

> 



(2.1) 



where d. . is the flexibility coefficient defined as the 
displacement of mass M i" due to unit load applied at 
mass "j M . 

Assume that the displacement at each mass point 
has the following form 

x^t) = x_ L Sin oD t (2.2) 

So that 

x^t) = -u) 2 x i Sin to t (2.3) 

where is the half amplitude of vibration of mass w i" 
and ^ is the natural frequency of the structural system. 
Substitute Eq. (2.2) and Eq. (2.3) in Eq. (2.1), and 
cancel out the common term, SincOt. From this, we 
obtain 





A l 








XI 

i 








ro 1 d 12 . . . 


. . m n d2n 














I ' > 








m l d ln ' ' ' 


n nn 





(2.4) 



Where cD is the natural frequency 

m^ is the lumped mass at point i 
d^j is the displacement of mass i due to unit load 
at mass j (i.e., an element from the flexi- 
bility matrix) 
x^ is the ordinate of the natural mode at mass i 

By assigning arbitrary values of Xj/ 1 ^ i n Eq. (2.4) as 

the first trial, and defining the product of the matrices 

(1 ) 

m i^ij and x i as Yi * then Eq. (2.4) becomes 



(1) 
1 

(1) 



> - 



(1) 



a 



where matrix D 



(1) 



x l 



(1) 



D 



> - 



(1) 



yi 



(i) 



y 2 



(1) 



represents the matrix 



m i d ij 



(2.5) 



25 



Modify the two matrices j I and I y<'*H as follows 



so that X, and Y, are normalized to 1.0 

(1) 



X.<» , ^ 



1 "V 11 



(2.6) 



Vf (1 ' = ^ (2.7, 



y l 



(1) 



for i ^ 1, 2, . . . , n. 

(1) 



If j *i j does not closely agree withjY 1 [L) \ for all 

(v (1) 



values of i, then the values of j Y^ 



are substituted 



in matrix | Xj^ 



(1)1 



of Eq. (2.4) to compute |Y ( 2 H . 



The routine continues until a close agreement between 
jy^n-Dj and) Yi (n M is reached. Thus Y x (n) represents 
the ordinates of the fundamental mode at each lumped mass. 
The fundamental frequency is obtained by Eq. (2.8) 

03 * Yl^ 

L ° 1 " ( n ) (2.8) 

The procedure for obtaining higher modes is derived as 
follows : 

Let x im and x in denote the ordinates of mass i at 
mode m and n respectively. Then the normalization equation 
requires that 



26 



N 



x. x, m - 0 
ira In i 



Si 



* n 



(2.9) 



i=l 



Assume that we have obtained the first k-1 modes and seek 

the k th mode. Eq. (2.9) can be written for each of 

the k-1 modes letting m take on the values 1,2, ... , k-1 
and n s k. Further, let 

r = x A m 

** " *im 

so that each known mode shape is scaled to 



r, = 1.0 

1m 



Transpose all terms having i > k-1 to the right hand side 
in each of the k-1 equations. The results of these 
operations can be written in matrix form as follows: 



r ll m l 
r i2 m i 



r k-l,l m k-l 
r k-1.2Vl 



f x lk 
i x 2k 



r i,k-iV** r k-i,k-i m k-i 



c k-l.k 



r kl m k 
r k2 m k 



r N1 m N 
r N2 m N 



r k,k-l m k»" r NN ra N 



x kk 

x kU,k 



x Nk 
(2.10) 



27 



Eq. (2.10) can be reduced to 





1 K 

x 2k 




X l*L- 
KK. 

x k l.k 






"b1 < 

J 


x k-l.k 


> m .[o] < 


x Nk 

■* -J 


> 


(2.11) 



where matrices [b] and [c] represent the corresponding 
coefficient matrices of Eq. (2.10). Pre-multiply Eq. (2.11) 
by [b!- 1 , the inverse of [B "] hence 



"Ik 
x 2k 



B 

s. J 



'kk 

: k+l.k 



Let BC 



Vl.k 



= .r B] .i [c 



(2.12) 



c Nk 



Then the natural mode for the k tn - 



mode can be written as 



c lk 
C 2k 



l Nk 



1 



r i 
1 00 J 



lk 



'2 k 



> 



x Nk 



lk 

C 2k 



, x Nk 



(2.13) 



28 

where [Z*] is a N x (k-1) null matrix, and [i] is a 

(N-K 1) x (N-k 1) unit matrix and [s k ] is defined as the 

sweeping matrix which sweeps out the lower modes. 

Substituting Eq. (2.13) in Eq. (2.5), we have 
the general expression for iterating all the higher modes: 



v lk 
C 2k 



'lk 
c 2k 



Nk 



l Nk 



(2.14) 



The iterative procedures for the higher modes are essen- 
tially the same as those of the fundamentals. But the 
result will be less accurate due to the "round- off" errors 
common to all numerical procedures. 

The Escalator Metho H (17) 

The escalator method does not involve iteration, 
nor does it involve the evaluation of determinants. It 
uses only the successive approximation procedure. For 
example, if we want to find the solution of a fourth 
order characteristic equation such as follows: 



'll-X a 12 



a 



13 



'14 



x l 



a 21 



a 31 



a 41 



a 22 -\ a 23 



a 24 



a 32 



a 42 



»33 -\ a 34 



a 43 



a 44 -X 



*2 
x 3 
1*4 J 



>=0 (2.15) 



where \ - 

we can obtain the value of X and its corresponding 
natural modes through Eq. (2.16) 



3 

V P 3r /(A 3r- X) = 3 44 " ^ 



(2.16) 



r 1 



in which P 3r = (a 14 ) x lr + (a 24 ) x 2r + (a 34 ) x 3r (2.17) 
r = 1, 2 P 3. 

In Eq. (2.17) \ 3r and x ir are the roots and the normal 
modes of the third order characteristic equation as 
defined by Eq. (2.18) . 



J ll 



J 21 



'31 



a 12 



j 22 



a 32 



a 13l 



! 23 



f 33 



x 2 L - 0 



i, x 3 



(2.18) 



The values of \ of the forth order equation as expressed 
in Eq. (2.13) are separated by those of the preceeding 



third order equation, namely by X 3^X321 andX 33 . The 
same procedure can be extended to a n th order equation. 
However, the application of this method to the general 
case always involves finding the roots of higher order 
characteristics equations. Because that operation is 
not always straightforward, the Stodola-Vianello pro- 
cedure is preferred. For the purpose of this disserta- 
tion, the Stodola-Vianello method has been adopted for 
use in computing natural frequencies. 



CHAPTER III 



REVIEW OF THE AVAILABLE NUMERICAL PROCEDURES FOR 
COMPUTING THE DYNAMIC RESPONSE OF 
CONCENTRATED- MASS SYSTEMS 

3 . 1 Introduction 

The equation of motion of a concentrated-mass 
system with multiple degrees of freedom contains a 
set of second order simultaneous differential equations 
with constant coefficients. Because of the complexity 
involved in its mathematical manipulation, the analytical 
solution to this type of problem is seldom obtained. The 
alternate approach is provided by numerical solutions 
which yield approximate values of the true solution. 
Their accuracy depends on the magnitude of the time 
increment and the given initial conditions. 

The numerical methods applicable in the struc- 
tural dynamics field cannot be classified rigidly into 
different categories, because many of them are inter- 
related. But all the numerical solutions can be 
expressed in two forms: 



31 



32 

a) given initial conditions and exact derivatives 
of all orders, or 

b) given initial conditions and differences of 
all orders. 

However, some of the numerical methods are 
manipulated strictly on a numerical basis and require 
no physical concept of the structural system once that 
system has been used to formulate the equations. 

In the subsequent sections, some of the more 
popular approaches to the solution of first and second 
order ordinary differential equations are illustrated. 
Their applicabilities to structural dynamics problems 
are judged so that the proposed solutions presented in 
Chapter IV can be placed in proper perspective. 

3.2. Successive Approximation Methods (26) 

The earliest solution of this type was due 
to Euler (26). Consider the first order equation (3.1). 

y - & - f(*.y) (3.i) 




Y 

i 



33 



y*ftx) 
















y, 








h 


h 

>»- 






= A x 


= A X 


1 



Fig. 3. EULER'S SUCCESSIVE APPROXIMATION METHOD 



v 



Then, from Fig. 3 we assume that any small change 
in y» flY. corresponding to a small change in 
x, Ax, can be expressed as 

A y = Ihx) a x 

in which the slope y* is assumed to be constant 
between x^ and x^_^. Thus, 



(3.2) 



y l s y o + 



dy_ 

dx, 



(3.3) 



\ I o 



where 

h m A x 

Eq. (3.3) can be put in a general form as 



34 

h i . 1, 2, * * ' , n (3.4 ) 

/i-1 

The accuracy of Eq. (3.4) will depend on the value of h. 
The whole process is reasonably fast but the solution 
obtained diverges from the true solution as x increases 
from the initial value. Therefore, it is not suitable 
for use in structural dynamic problems. 

3.3. Series- Expansion Method 

All the well-defined functions can be expanded 
by the Taylor series as shown in Eq. (3.5) 

y ( x + h ) = y ( x ) 4 y • ( x )Jl_ + y" ( x + • • • + R n 

(3.5) 

where R is the remainder and is shown in Eq. (3.6) 
n 

R n .i?f°<£) 0.6) 

where x-h < £ < x 

The conditions imposed on Eq. (3.5) require that 

the function y(x) should have all order of derivatives 

and that when n— > oo , the remainder R should approach 

n 

zero . 

Based on a series expansion, Milne (16) proposed 
a method which requires four consecutive initial values 
of y to start the process and utilizes two equations 



35 



called the "predictor" and the "corrector." The pre- 
dictor is given by Eq. (3.7) and the corrector by 
Eq. (3.8) 

y n + l - Vn-3 + &l*' n " vLl + 2y n _ 2 ) + 

(3.7) 

W " V n+1 +§(Y n+1 + 4y; + y._ l} . I_ h 5 y (5) 

(3.8) 

In each step of computation, the values obtained from 
Eq. (3.8) should agree closely, otherwise the magnitude 
of h should be reduced. However, four consecutive 
initial values are not frequently available in 
structural dynamics problems. In seeking a general 
approach, solutions of a similar kind are not considered 

3.4 Finite Difference Method 

The numerical solutions of first order equations 
that use finite differences are numerous. For the 
purpose of illustration, only one method will be 
discussed, namely the Nystrom method (5), which is 
represented by Eq. (3.9) and Eq. (3.10). 



Y n + 1 * Vl + h 



2y n + £( V 2 + V 3 + v 4 )y 

n 

(3.9) 



y , s y , + h 

n+l n-1 



37 6 
3730 V f n +l + 



(3.10) 

where V is defined as the n+l tn order backward 

difference and is expressed mathematically by Eq. (3.11) 

_ n+l n n . . 

v y n - V y n - v y n _i (3.U) 

Eq. (3.9) is a predictor equation, whereas Eq. (3.10) 
is a corrector equation. 

Nystrom's method needs five starting values, 
which place it in the same disadvantageous position as 
Milne's method. The finite difference method is based 
on series expansion concepts. 

3.5. Runge-Kutta Method 

The Runge-Kutta method (24) is also based on 
a series expansion. However, the solution can be 
started without the use of any initial values beyond 
those prescribed. The choice of h depends on the 
accuracy and the speed of computation, and h remains 
constant throughout the range of computation. For 
the first order differential equation the increment 
in y can be represented by Eq. (3.12) 



y = _(kj_ + k 2 + k 3 + k 4 ) (3.12) 



37 



where 

k l = f(x o' V )h (3-13) 

! k 2 - f ( x 0 + §■ y 0 + ( 3 - 14 ) 

k 3 = f ( x o + I' y o + (3.15) 
k 4 = f(x Q + h, y Q + k 3 ) (3.16) 

Obviously the procedure is very laborious and 
checking is achieved by a second computation. This 
method, however, possesses a high degree of accuracy. 
Further, it can be extended to second order differential 
equations . 

3.6. Acceleration Methods 

The equation of motion of a single mass m can 
be shown as 

I mx(t n ) = F(t n ) - R(x) (3.17) 

where 

x(t n ) = acceleration of mass m at time t n 

F(t n ) m forcing function at time t 

R(x) = elastic resting force due to displacement 

From Eq. (3.17) the acceleration x(t n ) can be written 
as 

S(t„) ■ F( V - R n (3 . 18) 

m 



38 

The analytical solutions for displacement and velocity 
at the end of a time interval A t n are 



<u ) f tn+1 
= x(t J + I 



x(t n+1 ) = x(t n ) + J x(t)dt (3.19) 

t. 



and 

l(t n4 l> - *<*„> + 4 «*n> at „ 



4- 



t n 



j x(t)dt 



dt (3.20) 



where 



At - t , - t 
" n n+1 n 



If the acceleration- time relationship is known in the 
time interval t , Eqs. (3.19) and (3.20) can be inte- 
grated directly. However, the expression of x ( -t^ ) 
cannot, in general, be integrated conveniently. In order 
to solve Eq. (3.17) by the iterative method, the 
acceleration x(t) must be assumed in the interval t n , 
and Eq. (3.19) is used to find the trial value of x, 
which is then substituted in Eq. (3.18) to obtain the 
acceleration. The comparison of the assumed accelera- 
tion and the computed acceleration provides a convergence 
criterion for this iterative procedure. 

In the succeeding discussions, various assump- 
tions about acceleration will be studied. The 
following notations are adopted for simplifying equations. 



39 



'n+1 

dx] 
dt t 



s X 



n+1 



n+1 



dt 2 M 



n+1 



x(t 



n+1 



n+1 



3.6.1. Constant Acceleration Method (24) 



The initial values of x 



x", and x at t = t 
n n n 



n 



are known. By selecting a sufficiently small time 

interval, A t , one may assume that the acceleration 

as shown in Fig. 4 remains constant with t and is 
3 n 

equal to x n . 




Fig. 4. CONSTANT ACCELERATION METHOD 



Based on this assumption, Eqs. (3.19) and (3.20) give 

the values of x , and x , at the end of the interval, 
n+1 n+1 

and are expressed as 



40 



X n+1 " X n + X n At + £ x n At n ( 3 - 21 > 

x , x + x &t (3.22) 
n+1 n n n 



The succeeding values of x n+ 2 anc * x n +2 are obtained by 
substituting x n+ ^ and x n+ ^ into Eq. (3.18) to get 
the approximation of x n+ ^» which is again used in 
Eqs. (3.21) and (3.22) to extrapolate displacement and 
velocity for the next time interval. Although this 
version does not require iteration, it is rather 
inaccurate (to the second order of h) and is not used 
in practice. 

Timoshenko (36) modified the assumption of this 
method by letting the acceleration in the time interval, 
t , be taken as the average of the beginning and end 
accelerations. Thus, 

Thereby, the equations for the displacement and velocity 
become 

x i a x + x h + I h 2 (x n 4- ft, ) (3.23) 
n+1 n n 4 n n+I 

x . « x + I (x + x , )h (3.24) 
n+1 n 2 n n+1 ' 



41 

Since the unknown >^ n+ 1 i s shown on the right 
hand side of the equal sign, the solution can be obtained 
only by an iterative method in which Eqs. (3.18), (3.13), 
and (3.24) are used. The accuracy of this method is of 
the third order of h. 

3.6.2. Linear- Acceleration Method ( 24 ) 

The acceleration in the time interval, t n is as- 
sumed to have a linear variation as shown in Fig. 5 and 
Eq. (3.25) 




Fig. 5. LINEAR ACCELERATION METHOD 



42 



*(t) = x n+ *n»l " * n (t - t n ) (3.25) 



where t n < t < t n + x 

Substitute Eq. (3.25) into Eqs. (3.19) and (3.20), to 
obtain displacement and velocity, respectively. Thus, 

' h A h 2 (0 " . " ) (3.26) 
*n + l = x n + x n + ™ (2x n + x n + l > 

x n+ l ■ x n + jj (*n + *n + l> < 3 - 27 ) 

Again, iteration is required because x n+1 appears on the 
right hand side of Eq. (3.15). The accuracy of this method 
is of the third 'order of h. 

3.6.3 Parabolic Acceleration Method ( 24 ) 

The acceleration in the time interval, & t, is as- 
sumed to behave like a parabolic function of time t as 
shown in Fig. 6. 




43 



n-1 n n+1 n+2 



Fig. 6. PARABOLIC ACCELERATION METHOD 
Mathematically this can be written as 

x n+l a x n + k l h * k 2 h2 (3.28) 

There are two coefficients, kj_ and k 2 » to be de- 
termined; therefore, one more initial condition is 
necessary for starting the solutions. The displacement 
and the velocity are expressed by Eqs. (3.29) and (3.30), 
respectively . 

2 

x n+l s x n + *nh + — (-* n -2 + 10* n -l + 3x ) (3.29) 

24 

x n+1 = x n + £L_ (-5 n-1 + 8x n + 5x n+1 ) (3.30) 

The iterative procedure is also needed in this method, and 
the accuracy is of the fourth order of h. 

3.6.4 The (3 -Method (22) 

A very versatile procedure in the category of ac- 



44 

celeration methods was developed by Newmark (22). A 
parameter " /3 " is inserted in the displacement function to 
increase the efficiency of an iterative method. The dis- 
placement and the velocity, respectively, are represented 
by 

x n + l * x n + V + <i " P > x n h2 + I 3 x 'n+l h2 ( 3 - 31 ^ 

' ' " h " h i n 11 \ 

x n+1 - x n + x n ^ + x n+1 - (3.32) 

Eqs. (3.31), (3.32), and (3.18) form the basic equations 

for the iterative procedure. For (b = i , the method is 

6 1 

equivalent to linear acceleration method; (i s - corres- 
ponds to Timoshanko's method; and /3 = — corresponds to a 

8 

step function with a uniform value equal to the initial 
value at the first half of the time interval and a uniform 
value equal to the final value at the second half of the 
time interval. Fig. 7 shows graphically the equivalence of 
each (3 -value as related to each acceleration method. 



45 




Fig. 7. CONSISTENT VARIATIONS OF ACCELERATION 
IN A TIME INTERVAL 

(Source: Fig. 3. Newmark (22)) 

The advantage of the ^-method is that the time increment h 
and the value of (b are inter-related. Therefore, as long 
as the stability criterion is satisfied, one may choose 
the /3 to fit the time increment h. The |3 = — procedure 
is strongly recommended because it is not restricted by 
the instability criterion and it is very convenient in 
analyzing a multi-degree freedom system. The accuracy of 
this method depends on the choice of [3 and h. 

3.7. General Remarks on the Numerical Integration Methods 
3.7.1 Errors 

Two kinds of errors exist in the numerical integra- 
tion methods, namely the truncational error and the round- 
off error. The former is the result of eliminating the 
higher order derivatives of differences in the recurrance 



46 

equations, and the latter is purely due to mathematical 
manipulation. The truncational error can be reduced by 
retaining the higher order terms and decreasing the time 
interval h, which in turn, increases the amount of labor in 
the process of computation. It is practical to predetermine 
the accuracy to be desired in the computation so that the 
error in the result can be predicted. The round-off error 
is mechanical and is controllable. By retaining a suffi- 
cient number of digits in each stage of computation, the 
round-off error can be minimized to a certain degree. 

The truncational errors which exist in the dis- 
placement and velocity will propagate in the process of 
computation but do not have magnifying effects. The same 
error in the acceleration will have some, but only slight, 
effects on the later stage. 

3.7.2 Stabili ty 

This criterion applies to those method using an 
iterative scheme. There is a close relationship between 
the time increment h and the limit of stability. If the 
value of h exceeds the limit of stability, the numerical 
solution will oscillate without bound; hence, no solution 
can be reached. Detailed discussions of stability problems 
have been treated by Newmark (20), (21) who also provided 
a table (22) of convergence and stability limits for 
various values of h and . 



47 

3.7.3 Choice of Numerical Methods 

The accuracy, the computing tolls and the personal 
perference together with the prescribed initial conditions 
given in the problem will determine the most suitable method 
to be used in numerical integration. If starting values of 
a problem need to be computed, they should be done with high 
accuracy. Runge-K.utta and Taylor series procedures repre- 
sent two good methods for this purpose. If the Modified- 
Eulerfs method is used, the time interval should be kept 
very small. 

Checking is another major item that needs special 
consideration. In the iterative method, the error committed 
in one stage does not influence the final solution; therefore, 
it does not require special attention, Milne and Adam's 
methods have their own checking equations that prevent any 
accidental mistakes. For methods without built-in checking 
equations, the mistakes can be detected only by using higher 
order differences. Any irregularity in the higher order 
gives a warning sign of existing error, and the error can 
be traced if the difference table is properly tabulated. 

3.7.4 Final Comment 



Most of the procedures outlined in this chapter have 
used in the past for the solution of structural dynamic 



48 



Probes. Of those listed, the Runge-Kutta method and 
the Newmarx |t -method seem to have the most genera! and 
confident application. However, in all of these methods 
the influence of the various modes is included in the 
solutions in an implicit manner so that it is impossible 
to separate and evaluate the contribution made by each 
"Ode. In the present study it was desired to identify the 
various contributions so that the influence of each could 
be measured. Therefore, the procedure outlined in the fol- 
lowing chapter has been adopted for use. In the method to 
be described the various modes are treated explicitly 



CHAPTER IV 



THEORY OF THE DYNAMIC RESPONSE OF 
CONCENTRATED- MASS SYSTEMS 



4 • 1 Free Vi b ration of a Concentrated-Mass System 

As shown in Chapter II, the characteristic equations 
for such a mathematical model can be written as 



0) 



I X 2 



> 



m l d ll 



m l d 12 



m l d in 



rn d,- _ 
n in 



, m d 
n mn 



X l ^ 



X 2 



"n 



(4.1) 



Eq. (4.1) is usually solved by a matrix iterative method, 
such as the Stodola-Vianello procedure as described in 
Chapter II. Thus if such a method is employed, Eq. (4.1) 
will yield n natural frequencies as shown in Eq. (4.2) 



0 



I 

CO n 



(4.2) 



The corresponding natural modes are given by Eq. (4.3) 



49 



X. 
. imj 



11 

: 21 



X , 
„ nl 



X 



2n 



50 



(4.3) 



where x im denotes the m th natural mode at mass point "i« 



Let 

X im - m i X. a) 2 



(4.4) 



The ^jm is defined as the characteristic load at mass point 
j for the m mode. Let the characteristic loading corres- 
ponding to the n mode be applied to the structure. Then 
let the structure undergo a virtual displacement corres- 
ponding to the m th mode. The virtual work done in this 
case is shown to be equal to that of the characteristic 
loading of m th mode acting on the structure while the 
structure is subjected to a virtual displacement corres- 
ponding to the n th mode. To express this in mathematical 
form, we write 



N 



i=l 



N 



im in 



^ in X im 



(4.5) 



i-1 



Substitute Eq. (4.4) in Eq. (4.5), to obtain 



51 



N 

2 



m i l^n x im x jm) " ^ m i ( «V x im X jmM (4.6) 

iml i«l 

or, n 

^m 2 -^n 2 ) ^ m i X im X jm ' 0 

Since 



C«) »« u) m if ui m (4.7) 



then ^ 



m. X. X. = 0 
i im in 



1 = 1 

Eq. (4.7) is called the orthogonality condition. If 



n ■ m 

then 



<*) - u) « 0 
m n 



Hence 

N 



2^ m i X im 2 = any arbitrary constant (4.8) 



i-1 



If we let the arbitrary constant be taken as one, 
then the characteristic shape of the m tn mode is said to 



52 

have been normalized. Letting the normalized charac- 
teristic shape be denoted by <i j_ m , the orthogonality 
condition can be written as 

N t=o if m » n 

in ( (4.9) 
i«l 1=1 if m ■ n 

The normalized characteristic load of m^h mode at mass 
point i is defined as 

L im sm i$i^ m 2 < 4 - 10 ) 

If the concentrated-mass system is subjected to a 
set of static loads, Oj_ , at each mass point i, it has been 
shown (24) that the static load at mass point i can be 
represented by the sum taken over all the modes of the product 
of the characteristic load L. and P . Or, mathematically, 
this can be written as 

N 

Qi - ^ P m L im (4.11) 

where P m is defined as the participation factor for the 
n" t '" 1 mode. This is expressed as 



N 

' Q i $ im 



53 



Similarly, the static displacement of mass point i due 
to static loads Q i « s may be expressed as 
N 

4.2. Pynarnic^sEonse_of_a_ Con centrated-Mass S v stem 
Subjected to Dynamic Loading 

Let a concentrated-mass system be subjected to a 
set of dynamic forces, O^t), Q 2 (t), ...,Q n (t), as shown 
in Fig. 8. 



ra- 



ni. 



m 



rn 



N 



o x (t) 



Q 2 (t) 
Q 3 (t) 



Q N (t) 



77wr 



Fig. 8. CONCENTRATED-MASS SYSTEM SUBJECTED 
TO DYNAMIC FORCES 



The total displacement response of mass point i at time 
t would be the superposition of the displacement due to 
each individual load Q 1 (t)'s effect and that due to 
initial motion of the system. 



54 

Denote the dynamic displacement of mass i to 
be x i T (t). 
Then 

*i T (t) = x i °(t) + x.(t) (4.14) 

where x j .°(t) and x^t) represent the dynamic displacement 
of mass i due to initial motion of the system and the 
dynamic forces, respectively. 

To simplify the derivation for Eq. (4.14), a 
two-mass system as shown in Fig. 9 is used. 

m a— Qi(t) 



rri2 



Q?(t) 



777777" 



Fig. 9. A TWO-MASS SYSTEM 



Assume that the dynamic forces, Q 1 (t) and Q 2 (t) can be 
expressed by Eq. (4.15) 



2 

Qi(t) « q A f(t) . V p m L . m f(t) 

m=l i=l,2 



(4.15) 



where q is the maximum magnitude of Q i ( t) , and f(t) 
is a function of time only. 



55 



Assume that the dynamic displacements of mass i due to 
dynamic forces and initial motions may be expressed by 
Eqs. (4.16) and (4 . 17 ), respectively , 



x °(t) b V D ©(t) $ 

1 c—t m 

m«l 

2 



im 



x^t) 



E 



(4.16) 



(4.17) 



m«l 



where D m (t) and D m °(t) are functions of time, 



m 



Substitute Eqs, (4.16) and (4.17) in Eq. (4.14). The 
total displacement x i T (t) becomes 

2 



*i T (t) 



2 



m=l 



(4.18) 



m-1 



The displacement x i T (t) could also be expressed in terms 
of dynamic and inertia forces as shown in Eqs. (4.19) 
and (4.20), 



T 



x x (t)= Q 1 (t)-m 1 x 1 1 (t) 



x|(t). 



Q 1 (t)-m 1 x 1 T (t) 



d n + 



d + 
21 



Q 2 (t)-m 2 x 2 T( t ) 



Q (t)-m 2 x 2 T (t) 



u 12 
(4.19) 



u 22 
(4.20) 



where d. j • s are the flexibility factors 



56 



Substitute Eq. (4.18) in Eqs. (4.19) and (4.20), 
and regroup terms. The following equations are obtained: 



f(t)P m L 



m^l 



lm-» A < t ) $ lm ] d, , + [ f ( t ) P m L 2m - m2 D m ( t ) P m # 2 J d 



12 



■[ m l£im d n +m 2 £ 2m d 



12 



B m°(t) - D m (t)f im . D m o (t )$ 1 I . 0 



2 , 



E([ f(t)p « L i»V«< t ) p «Li B ]d 21 + 

mel 



f(t)P m L 2 ra Vm' t ' P m« 2m j d 22 



"if U^l^a-^J D o, t , . D lm (t)P m f „-D »(t)f , 1 . 0 



* y * 2mj 
(4.21) 



Eq. (4.21) can be reduced to 



in 



m 



D °m(t) 

L u> 2 
m 



+ D °(t) 
m 



- 0 

(4.22) 



It can be shown (24) that 



m 



cO 2 
m 



+ D °(t) = 0 



(4.23) 



Theref ore , 

5 m (t > +a) / D m(t) 



^m 2 f(t) 



(4.24) 



57 

The solution for Eq. (4.22) is 



Vl')' 0 , 0 ".) Cc^t-to) + ^Ihl Sin^ m (t-t 0 ) 

(4.25) 

where 



m=l 



2 

m=l 

Here, x A (t 0 ) and x^tj are the initial displacement and 
velocity of mass i at time t , respectively. 
The solution for Eq. (4.24) is 

D m U) = f (*J Sin^t-t'jdt' (4.28) 

t 



To extend Eqs. (4.26), (4.27), and (4.28) to an N-mass 
system, we simply change the indices 2 over all summation 
signs to N. Then, the total dynamic displacement of mass i 
for an N-mass system becomes 



58 

N" 

Xi T (t) - V[D m O(t 0 )Co^ m (t-tJ + ^l^ Sin^t-tj] $. m 
m-1 L m 



+ ) ^im^m ' 
mil 



f(t»)Sin^ m (t-t')dt» (4.29) 



*0 



Eq. (4.29) is valid only when all the Q (t)"s have the same 

j 

function of f(t). Thus for dynamic forces which do not 
have a common f(t), the seond term on the right side of 
Eq. (4.29) should be computed separately for each load 
Q j ( t ) , which is expressed as 

Q.(t) = q.f.(t) (4.30) 
J J J 

Then for each Qj(t), the participation factor P m is changed 
to 

q j $ jm , 

P m = *r~ 4 - 31 

m 2 

m 

Denote x— (t) to be the dynamic displacement of 

mass i subjected to load Q^(t); the expression for. x--(t) 
could be written as 



59 



N 



x xj (t) 



m = l 



q j £ im£jm f { 



j (tMSin^ fn (t-t 1 )df 



(4.32) 



Thus, the dynamic displacement of mass i due to all the 
dynamic loads has the following expression: 
N N + 



x A (t) « 



V q J ^ im^jm 



m=l 



m 



f j (f)Sin<O m (t-t 0 )dt' 



J 



(4.33) 



iherefore, the total displacement of mass i due to 

Qj(t)'s is obtained as 
N 

N N - _ t 



/ 



m 



j=l m»l 



a) 



m 



0 (4.34) 
This procedure will be used to study several structures. 



CHAPTER V 



THE ADOPTED NUMERICAL SOLUTION FOR CALCULATING 
THE DYNAMIC RESPONSE OF TRUSSED TOWERS 

The adopted method is based on the assumption that 
the natural frequencies and the normal modes of the steel 
tower can be determined by its equivalent concentrated-mass 
structural system, and the wind forces which are acting at 
the mass points can be represented either analytically or 
quantitatively by any statistical method. 

Further, the procedure assumes that all of the 

natural frequencies and modes of vibration of the structure 

are known. However, numerical examples presented in Chapter 

indicate that quite accurate results may be obtained when 

only the first few modes are included in the analysis for 

real towers of practical proportions subjected to natural 

winds. Therefore, the procedure should be adequate if the 

first few modes of a real tower can be predicted by analytic 

means. The experiments described in detail in Chapter VI 

offer a valuable confirmation that the first two modes of 

vibration can be predicted with reliability by the idealized 

concentrated-mass system provided the base flexibility can 

be evaluated adequately. Although it was not possible to 

check the higher modes of the test tower because of the 

60 



61 

limited capabilities of the exciting equipment, 
no evidence developed in the tests conducted that would 
lead us to question our ability to compute with sufficient 
accuracy the third or fourth mode of a tower structure. 
Therefore, the procedures outlined in this chapter are 
offered as an acceptable approach to the computation of 
the dynamic response of real towers. 

5,1. Assumptions for the Forcing Function Qj(t ) 

The wind forces on the projected areas of the 
tower members are assumed to act on the concentrated 
masses that represent the lumping of the particular members 
considered. The wind force on non- structural platforms, 
equipment, etc., is also assumed to act on the nearest 
concentrated masses by appropriate equilibrium calcu- 
lations. Hence, the equation for expressing the wind force 
at mass point j can be written as 

Q.(t) = C A P (t) (5.1) 
J d j j 

where 

C, = drag coefficient of the projected wind areas 
d 

assigned to mass-point j 
A. = the lumped projected wind areas assigned to mass- 
point j, taken normal to the wind 
p.(t) = dynamic pressure of wind, expressed by Eq. (5.2) 



62 



P.(t) = 0.00256 VAtf 



(5.2) 



Here , 



V (t) = velocity of wind at mass-point j, in miles 
per hour. Eq. (5.1) can be written as 



Thus, the wind force at each mass-point is directly pro- 
portional to the square of the wind velocity, Vj(t). 
Implicit in this equation is the assumption that the wind 
velocity changes used in the analysis occur simul- 
taneously over a sufficiently large area to affect equally 
all projected areas that contribute to the wind force con- 
centrated at a particular mass point. Further, there is 
the implied assumption that the dynamic pressure acting on 
a particular wind area responds immediately to changes in 
wind velocity. There is some aerodynamic evidence (6) 
that there is a definite time lag between the establishment 
of new wind velocities and the development of the corres-c 
ponding dynamic wind pressures. However, because the 
primary purpose of this study is to validate an analytical 
procedure, it will be assumed in the remaining discussion 
that the wind velocities quoted are the "effective" wind 



Q At) = Q V (t) 2 
J J j 



(5.3) 



where 



Qi = 0 o 00256 x C. x A. 
J d j 



(5.4) 



63 

velocities after corrections for the above effects have 
been made. However, the predictions of real tower 
responses may be modified when more data are available 
on the aerodynamics of wind pressure changes. 

Furthermore, we restrict the wind velocity, 
Vj(t) to be the velocity component normal to the as- 
sociated projected area of mass-point j. The displacement 
of mass-point j produced by the transverse component of 
wind at that point is assumed to be independent of that 
in the direction of V j ( t ) . 

The function of Vj(t) is assumed to be a piecewise 
continuous function such that it possesses all order of 
derivatives at any specific time t, and such that the 
numerical values of Vj(t) are also obtainable. 

5,2. The Deri v ation of Numerical Procedure 

The dynamic displacement of raass-point i due to 
all the wind forces Qj(t) was given by Eq. (4.36) as 



N N « x *s A 
x i( t) = f V- J * im * Jm 



f .(t')SincO m (t-f )dt» 
j «1 m^l m J t (5.5) 



'O 

where 

f (f) = V.(t') 2 
J 3 

However, fj(t) normally is a very complicated 
function of t; hence the evaluation of the integrand in 



64 

Eq. (5.5) cannot be accomplished by ordinary integrating 
techniques. Since the numerical values of fj(t) at any 
t are assumed to be known, the solution of x^(t) could be 
obtained numerically if the integrand may be expressed in 
terms of fj(t)'s. Let Ij(t) represent the integrand, or 



Ij(t) = f fj(t') Sinu) m (t-t') dt» 
t„ 



(5.6) 



Integrate by parts, letting 

u = f.(t') 
J 

.'. du = f ' j ( t • )dt ' 

and dv - Sinu) (t - t ' ) dt' 
r m 

/. I.(t)= £ilLlccScO m (t-t») 



m 



m 



f fi(t') 
- -JLi — .CcstO (t-t')dt' 

t t 03 m 



Again let u = f ' j(t' ) 

du = f " j( t ' )dt» 

and 



dv * — CpStO ( t-t» )dt ' 
m 



r 1 i 

V-/ CoscO _(t-tf)dt»« 



-1 n t 

— ~ 0 SintO ( t-t' ) 
(J ^ m 



m 



(5.7) 



*o 



Ij(t) 



fj(t) 



•Cos^ m (t-t' ) 



t t 
f-(t) 1 
.V y Sino3 (t-t») 
a) m 2 m 



t w , . 

liilisin^ (t-t') dt' 
3 m 2 m 

t_ 



By repeating the same technique, I,(t ) will have 
the following form 



ijtt). 



f lS t ' ? cos «) ( t-t' ft]* + 

m J t 

f.(t') it 
J_CoscO m ( t . t .) 



_ ■ «-o L m 
r *> 



f/ j (t) SincJ (t-ff 

«0 2 



+ 



L m 

1 J (t') coa«0 (t 
m 



liUllsin u3 (t-t')l 1 



cO 4 
m 



m 



t,^ 1 r f 5) (t') , 1 
+ -J — . — Sin cO (t-t» ) . 

oJ 6 m v 
m 



'0 

(5.8) 



Evaluate I,(t) between t and t, 
•J o 



I (t) = 

j 



'iua . £Vu) + f (4) (t) 

cO m m 3 cO m 5 



' f j(tp) . f(4)( to ) 



a) 



m 



^ m 3 
m 



£° s ^ (t-t») + 
m 



u) 2 
m 



^ °m 6 



J 



m 



(t-t 0 ) + 



(5.$) 



Substitute Eq. (5.9) in Eq. (5.5). The dynamic 
ment 

expressed as 



displacement due to forcing function f . (t) can now be 



N N 



j-1 m-ll o3 2 J 
m 



F j( t) 



(5.10) 



where 



66 



F (t) 
j 



I 



m 



m 



*j(t 0 ) - 



fj^o) f (4) (t 0 ) 



^ m 2 
m 



^ m 4 
m 



m 



u3 3 
m 



^m 5 



J (5.11) 

cos- m (t-t o) 



Sin^(t-t ) 
m o 



But to evaluate F (t) in Eq. (5.11) requires the 
derivatives of all orders of function f j( t ). and the 
process of differentiation for a wind function is by no 
means a simple one. Hence, the values of the derivatives 
must also be computed through a numerical method. 

Assume the wind loads are as described in Section 5.1, 
and the velocity of the wind, V A (t) , at any time t is known. 
Then, the function fj(t) for mass i can be obtained as 
follows: (see Fig. 10) 



67 



f.Ct ) 
J 



f( 2) 

J _f.(3) 



,-nMQJ / 



t t t t 

0 12 3 



Fig. 10. LAGRANGIAN INTERPOLATION POLYNOMIAL 



In the interval, t 



t^ there are four numerical values 



of fj(t) at equal time interval, h. The derivatives up 
to the third order for each function fj(t) can be evaluated 
by Eq. (5.12) derived from the Lagrangian interpolation 
polynomials . 



f<»>(t) = J51 h m Y A k f k (t) + R (5.12) 
J n n' Z_. 

k=o 

where m is the order of derivative, n is the number of 
known f k (t), A k is the multiplying constant (tables for 
A^ can be found in Ref. (1), and R is the error term 
which is expressed as 

R = hn+1 f ( n +D (£ ) (5.13) 



68 



where t„ < F C t 
o £3 N n 

In order to have an accurate result, the time 
interval, h, should always be less than 1.0. 

By substituting each value of f^( n )(t) in 
Eq. (5.11), the displacement x^(t) can be evaluated by 
Eq. (5.10). The displacements for the next time period 
are obtained by the same procedure with the displacement 
and velocity at the end of the previous period as the 
initial conditions. Numerical examples are given in 
Chapter VII. 

5.3. The Computer "rogram 

The numerical solution, as developed in the previous 
Section, is a straightforward procedure. However, the 
amount of work involved to compute each quantity in 
Eq. (5.10) is enormous even with a desk calculator. A 
computer program suitable for the IBM 709 has been 
written for making all routine computations. The complete 
listing of the program is shown in Appendix A. The 
definitions of variables and the names of all the sub- 
routines used in the program are explained in the fol- 
lowing sections. A flow chart is also provided. 



69 



.3.1. The Variables Defined in the FORTRAN Program 

A: coefficients used in numerical differentiation 

DI : dynamic factor of displacement due to initial 

motion D° m (t) 

DV: ratio of velocity of wind to time t 

ETA: displacement response due to impulsive load 

x (t) 
i 

ETI: displacement response due to initial motion, 
x° it t) 

ETALL: total displacement response, x T ^(t) 

H: time increment used in the numerical differ- .. 
entiation 
JL : mass point 
KCHECK: control number 
MCHECK: control number 

N: number of panels 
NCHECK: control number 

NM: number of modes to be used in the computation 
NR,R,N: register number of time increment for external 
force 

NRTEMP: temporary storage location for NR 
NTV( JL): number of loading input at mass point (JL) 



70 

NTVMAX : the largest numerical number of an input loading 

PHI: normal mode 

T: time 

TDD: time increment 

TEMD1: D °(tj 
m u 

TEMVI: D °(t ) 
m v o 

TEMP: product of TDD and RN 
TEMPV: wind loading 
TMASS: mass at each panel point 
V: velocity of wind 
VETA: velocity response due to external loading 
VETAL: velocity response due to external loading 
and initial motion 
VETALL: total velocity response 

VETI: velocity response due to initial motion 
W: natural frequencies 
Y: derivatives at time t 

•3.2. The Subroutines Used in the Program 

Subroutine INPUT: To read N, NM, TDD, NTV(JL) 

TMASS, W, PHI, T, V, and to 
write out the input information. 

Subroutine POWER: To compute the wind force by 

Eq. (5.3) at each mass point. 

Subroutine FORCE: To comput displacement and 

velocity responses to wind 
force according to Eq. (5.10). 



r 



71 



Subroutine GROJP: To compute the displacement and 

the velocity responses due to 
external force and initial 
motion. 

r 

Subroutine EXIST: To compute D m °(t 0 ) and D m °(t 0 ). 
Subroutine TOTAL: To compute total displacement 

and velocity responses. 
Subroutine MOVE: To compute displacement and 

velocity responses due to 

initial motion. 
Subroutine OUTPUT: To write out displacement and 

velocity response at time t. 
Subroutine NUMER: To computer derivatives of f(t) 

at each time t by Eq. (5.34). 
Subroutine PRODT: To compute a factorial of a number 

(i.e. m! ) . 



5.3.3. Flow Chart 



FLOW CHART 




INITIALIZE 
REGISTERS AND 
CONTROL NUMBERS 
NR 

MCHECKj JL 
NCHECK( JL 
JL 



3> 



COMPUTE SLOPES OF 
WIND FUNCTIONS 



CALL 
POWER 



CALL 
NUMBER 



CALL 
FORCE 





CALL 
MOVED 



CALL 
GROUP 



73 



JL = 


JL +L 


= 1 























CALL. 
TOTAL 




CHECK 
MCHECK( JL) 




= 3 




JL = JL + 1 



SET TEMPORARY 
STORAGE 
LOCATION FOR 
REGISTER NR 



CALL 
EXIST 



COMPUTE THE SLOPE 
OF THE WIND 
FUNCTION FOR THE 
NEXT TIME INTERVAL; 
AND INITIALIZE 
CONTROL NUMBERS: 
MCHECK(JL) = 2 
NCHECK(JL) = 2 





74 



INCREASE THE 
MAGNITUDE OF NR BY 
ONE TO COMPUTE 
RESPONSES FOR THE 
NEXT TIME INTERVAL; 
AND RESET ALL 
MCHECK(JL) TO 2 



TEMPT- 
T(I+1, JL 






RESTORE THE 




ORIGINAL 




VALUE OF NR; 


> 


AND SET 




MCHECK(JL) AN 




NCHECK(JL) TO 




2 















SET 


_ 


MCHECK( JL ) =3 


NCHECK( JL ) =+ 







SET 






MCHECK 




=2 


NCHECK 




= 2 




75 



5.4. Application of the Adopted Method to the Design 
of Steel Towers 



After the dynamic displacements of a mathematical 
model have been found by the adopted method, the equivalent 
static forces acting at the panel points of the real tower 
could be found by Eq. (5.14). 



W 

where 



(5.14) 



kj ■ stiffness matrix, inverse of flexibility matrix 
■ displacement matrix of the extreme position 



= matrix of a set of equivalent static forces acting 
at the panel points of a real tower. 
The static equivalent forces found by Eq. (5.14) can then be 
used to compute stresses in the structural members. 



CHAPTER VI 



EXPERIMENTAL DETERMINATION OF THE NATURAL 
FREQUENCIES OF A K-TRUSSED STEEL TOWER 

6.1. Purpose of the Experiment 

The dynamic displacements of a steel tower sub- 
jected to wind loading are predicted by the equations 
derived from a mathematical model which is a concentrated- 
mass system. The equation representing the displacement 
at any mass-point is a function of the wind loading, the 
natural frequencies, and the normal modes of the concentrated- 
mass system. However, the displacement of each mass-point 
represents the panel point of an actual steel tower only if 
the natural frequencies and the normal modes of the con- 
centrated-mass system closely correspond to those of the 
steel tower, Therefore, the testing of a moderate scale 
steel model was considered to be essential to verify the 
assumptions described in Chapter II. 

6.2.. The Test Model 

The test model is a fifteen foot tower having a 
rectangular cross section. Each of the four side trusses 



76 



77 

that form the tower are composed of six equal length 
panels that are braced with K-type diagonals. The trusses 
in the plane of vibration have a depth of 2 ft.-O in. while 
the trusses normal to the plane of induced vibration have 
a depth of 1 ft. 4-3/4 in. All diagonals were made from 
1 in. by 1 in. by 1/8 in. angles. All other members 
were fabricated from 2 in. by 2 in. by 3/16 in. angles. 
All connections were made by fillet welds. Complete de- 
tails of the model are given in Fig. 11. The total weight 
of the model was 370 lb. 

Because the oscillator that was used to excite the 
model was capable of producing a periodic force only along 
a vertical axis, the model tower had to be supported in a 
horizontal position. Some difficulty was experienced in 
providing a rigid support for the model. The braced frame 
shown in Fig. 11 was determined to be sufficiently stiff 
to provide reasonable support. Corrections were made for 
the flexibility of the support in the analysis. 

6.3. Instruments Used in the Experiment 

A Lazan LA-1 mechanical oscillator was used as a 
force generator. The principle of this oscillator is best 
described by Fig. 12 as shown below 



79 




Fig. 12. PRINCIPLE OF THE OSCILLATOR 

The masses which were attached to the two shafts ro- 
tating in opposite directions generate vertical force which 
can be expressed by Eq. (6.1) 

F = 2 Sin | N 2 (6.1 ) 

where 

F = _+ oscillating force output of the oscillator in 
pounds 

U = angle between eccentric weights in tenths of a degree 
N = speed of rotation of oscillator in revolution per 
second . 

The maximum capacity of the oscillator is +_1600 lb. at 1800 
rpm . 

The speed of the oscillator was determined by a 
Type 631-B stroboscope as manufactured by General Radio Co., 
Cambridge, Massachusetts. The range of the stroboscope can 
be v a ried from 600 rpm to 14400 rpm. 



81 



The signals of the velocity response of the free 
end of the test tower were shown on a screen of an oscil- 
loscope by means of a Sanborn linear velocity transducer 
(7LV4). The velocity transducer consists of two parts: 
a shielded cylindrical coil assembly, and a high induced 
force magnet. The relative motion of the magnet and the 
coil generates a voltage whose magnitude is proportional 
to the linear velocity and whose polarity indicates the 
direction of the motion. A photograph of the complete 
assembly of the instruments and the test tower is shown 
in Fig. 13. The schematic diagram of the electrical 
circuit is shown in Fig. 14. 



permanent 
magnet 



shielded 
cylindrical 

coi I - 



velocity of 
panel point 1 




oscilloscope 



Fig. 14. SCHEMATIC DIAGRAM OF THE TEST CIRCUIT 



82 



6.4. The Testing Procedure 

To find the resonant frequencies of the test 
tower, an alternating force was applied to its free end 
by the mechanical oscillator. The signals of the velocity 
of the free end were picked up by the velocity transducer 
and displayed on the screen of the oscilloscope. As long 
as the frequencies of the alternating force did not 
coincide with those of the test tower, the signals were 
quite small and indicated only the effects of background 
noise. However, the magnitudes of the signals were en- 
larged to a great extent whenever a natural frequency was 
matched. The driving frequencies that produced the 
greatest amplitudes were recorded as the natural fre- 
quencies of the model. 

Because the maximum speed of the oscillator 
is 3600 rpm, any resonance frequency higher than 60 
cycles per second could not be excited in the experiment. 
For the model tested, it was not possible to verify any 
but the fundamental frequency. By attaching additional 
weights, it was possible to lower all frequencies so 
that the first two could be verified. 



83 



6.5. The Theoretical Flexibility Factors and the Concen- 
trated Masses of the Test Tower 



The test tower as shown in Fig. 11 could be 
schematically represented by Fig. 15 




a) Line drawing of tower, 



m. 



m 4 ir»3 
• 9- 



m. 



-c 



b) Mathematical model. 



Fig. 15. THE IDEALIZED SYSTEM OF THE TEST TOWER 



For the purpose of computing the theoretical flexibility 
matrix of the model , it was assumed that the supports were 
absolutely rigid and that all joints were pin-connected. 
From this, each flexibility coefficient, d^ j , was cal- 
culated according to Eq. (6.2). 



N 



U ik u jk L k 



1J Z_j A k E 



(6.2) 



84 



where 



cL j = displacement of panel point i due to a unit 
load applied statically at panel point j. 
i, j = 1, 2, . . .6. 

u., = force in member k due to a unit load applied at 
panel point i. 

u = force in member k due to a unit load applied at 

jk 

panel point j . 
■ cross- sectional area of member k. 
= length of member k. 
m ■ number of members in the truss. 

E = modulus of elasticity of the material (29,000,000 psi) 
The resulting flexibility matrix, djj , was found to be: 



189 


.6583 


143. 


8582 


100. 


4875 


61.9745 


30 


.7465 


9 


.2320 


143 


.8582 


119. 


5748 


85. 


9184 


54.6895 


28 


.3182 


9 


.2320 


100 


.4875 


85. 


9184 


71. 


3484 


47.4054 


25 


.8898 


9 


.2320 


61 


.9745 


54. 


6895 


47. 


4054 


40.1205 


23 


.4516 


9 


.2320 


30 


.7465 


28. 


3182 


25. 


8898 


23.4615 


21 


.0334 


9 


.2320 


9 


.2320 


9, 


2320 


9. 


2320 


9.2320 


9 


.2320 


9 


.2320 



X 

10 -6 
in. /lb. 



The weights that were assumed to be concentrated at each 
panel point are as listed in Table 1. 



85 



cu 
< 

CU 

H 
< 

to 

to 
to 
< 

3E 
Q 
< 

H 

O 
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m 

§ 
D 



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CD 
C 
re 

a. 





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in 




in 














■ 


• 


• 


o 


o 


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o 






fH 


CM 


i — i 














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n 














• 


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sr 


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o 


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l. N 


, — ( 














CNi 




















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in 




in 




00 
















o 


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c 


J — . 

Q 




r-( 


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vD 




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in 




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co 


W CO 


rH 


4-> MM 


in 








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^ X 


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M <-> S 


• CM 


N CM 


• -H 


U 3 CD 


-t-> 


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W MS 


H X 


M X 


ft) x 


<D -P 


CD 


O 


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Q to 


> CM 


X CM 


Q rH 



rH C 

CO 

X I 

CM +-> 
<+-i 

X rH 

CM X 

-4 

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CO 



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in 



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1 


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c 




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1 


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1 


o co 
i 


rH 


i — 1 -P 

a) to 


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rH O 


rH O 







CM 

rH 
lO 

o 

X> 
CM 

rH 

in 
d 



CM 

o 

CO 

€<• 
vO 



CM 

rH 

in 
d 



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in 
co 



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o 
o 

d 



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X! 



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CD rH 
C 

ro a* 

CD 

Q © 



n 
:0; 

rH 

o 



in 

vO 
lO 



I — 1 



if) 

in 
rH 

d 



CM 

i0 

in 



c 

■H 
\ 
CM 

O -P 
CD C 
CO -H 
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C rH 

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a 

cn ro 
i/> a, 
ro 

2 @ 



86 



6.6. The Ex perimental Determin a tion of the Flexibility 
Factors 

The actual flexibility matrix was expected to 
differ from the theoretical matrix for two reasons: 1) the 
actual truss had welded rather than pin- connected joints; 
and 2) the supports were not absolutely rigid. The first 
factor would tend to make the real truss more stiff than the 
idealized truss. However, the details were such that the 
working lines nearly always intersected at a common point 
at each joint. Therefore, the assumption of pin-connected 
joints was closely realized and the joint effect was not 
very great. The fact that the supports were not rigid would 
allow the real truss to rotate as a rigid body so that a 
measured flexibility matrix should exhibit a systematic 
deviation from the theoretical matrix. 

Therefore, an experimental flexibility matrix for 
the model and its supports was obtained by measuring the 
deflection of each panel point as a fifty pound weight 
was moved successively from panel point to panel point. 
The resulting matrix (corrected to a one pound load) was 
found to be: 



87 



d if] 



256. 


00 


205. 


00 


142. 


50 


99. 


50 


52. 


50 


17.60 


205. 


00 


175. 


00 


130. 


50 


97. 


50 


51. 


00 


17.60 


142. 


50 


130. 


50 


110. 


00 


81. 


50 


43. 


50 


17.60 


99. 


50 


97. 


50 


81. 


50 


70. 


00 


43. 


00 


17J60 


52. 


50 


51 . 


00 


43. 


50 


43. 


oo : 


27. 


00 


17.60 


17. 


60 


17. 


60 


17. 


60 


17. 


60 


17. 


60 


17.60 



X 10 



-6 in 



6-7. The Rotation of the Support 



4. 



By comparing the theoretical and experimental 
matrices, one can readily see that the experimental coef- 
ficients exceed the theoretical coefficients by an amount 
that is proportional to the distance to the supports. This 
is more apparent from the matrix formed by subtracting this 
theoretical matrix from the experimental matrix. This 
matrix is the flexibility matrix due to support rotation 
and is given below: 

"66.30 61.10 42.0 37.50 11.80 8.40 
61.10 55.40 44.60 42.80 22.70 8.40 
42.00 44.60 38.70 34.10 17.60 8.40 
37.50 42.80 34.10 29.90 19.50 8.40 
11.80 22.70 17.60 19.50 6.0 8.40 
8.40 8.40 8.40 8.40 8.40-8.40 



l d ij- d ij 



-6in/ 



lb, 



88 



This matrix contains experimental errors that are parti- 
cularly large (on a percentage basis) for the points near 
the supports where deflections are of the same order of 
magnitude as the least division on the Ames dials used 
for the measurements. 

By usinj standard linear regression methods, the 
support rotational spring constant (which should be the 
same for all positions of the load) was determined for the 
load at each panel point. Because of the small deflections 
at panel points 5 and 6, those results were not considered 
in the analysis. The spring constants obtained for the 
load at points 1 through 4 are given in Table 2. 

Table 2. ROTATIONAL STIFFNESS OF SUPPORT 



1# load @ Panel Point K in in.-lb./rad. 

1 421.62 x 10 6 

2 433.33 x 10 6 

3 496.86 x 10 6 

4 430.33 x 10 6 



Because the result should have been the same for all 
positions of the load, the final estimate of the spring 
constant was taken as the average of the three values found 



89 



for the load at points 1, 2, and 4. The result for point 3 
deviated considerably from the other three values and was 
rejected as containing larger experimental errors. The 
spring constant used for further analytical work was 



K 



428.43 x 10 6 in.-lb./rad 



(6.3) 



This spring constant was used to calcuate a 
smoother flexibility matrix due to support rotations. 
That matrix is as follows: 

'75.492 62.91 50.328 37.764 25.164 12.582 
62.91 52.425 41.940 31.455 20.970 10.485 
50.328 41.94 33.552 25.164 16.776 8.388 
37.746 31.455 25.164 18.873 12.582 6.291 
25.164 20.97 16.776 12.582 8.388 4.194 
12.582 10.485 8.388 6.291 4.194 2.097 

By adding the smoothed flexibility matrix due to support 
rotations to the flexibility matrix caculated by idealized 
truss theory, the "true" flexibility matrix for the model 
was obtained. The resulting matrix is 



X 10" 6 in./lb, 



90 



265.1503 206.7682 150.8155 99.7205 55.9105 21.8140 

206.7682 171.9798 127.8584 86.1445 49.2882 19.7170 

150.8155 127.8584 104.9004 72.5694 42.6658 17.6200 

99.7205 86.1445 72.5690 58.9935 36.0435 15.5230 

55.9105 49.2882 42.6658 36.0435 29.4214 13.4260 

21.8140 19.7170 17.6200 15.5203 13.4260 11.3290 



X 

10" 6 
in. /lb 



This matrix is quite similar to the experimental matrix and 
could be thought of as a smoothed version of that matrix. 
This matrix was used with the concentrated weights listed 
in Table 1 to calculate the natural frequencies of the 
model by the Stodola-Vianello procedure. 

It is recognized that the support spring constant 
that was determined experimentally is a static spring con- 
stant only. Under dynamic conditions, its effective value 
could change. However, its value should change radically 
only if the tower began to vibrate at a frequency that was 
resonant with the natural frequency of the support system. 
Preliminary calculations indicated that the natural frequency 
of the support is greater than the highest speed of the 
oscillator used. Further, there was no indication of this 
difficulty during the tests although the supporting structure 
did contribute a considerable amount of noise when the model 
was excited in its second mode. 



91 

6.8. Test Result s 

In the first test, the only non-structural mass 

used was the oscillator which was placed at panel No. 1 

] 3 
and fastened by four - t in. steel bolts on a _ in. steel 

4 8 
plate as shown in Fig. 11. The concentrated masses at 

each panel point are tabulated in Table 3. 

The analytical and experimental natural frequencies 
of the test tower are shown in Table 4. 

Since the oscillator could produce an alternating 
force with maximum frequency of 377.0 rad./sec. (3600 rpm ) , 
the first test could not yield the desired second mode which 
was found analytically as 387.05 rad./sec. In order to test 
the second mode of the test tower, the masses at each panel 
point had to be increased. This was accomplished by adding 
extra masses at appropriate joints. A trial test was con- 
ducted by placing lead weights of approximately one hundred 
pounds each clamped rigidly at panel point 2 through 6, 
respectively. (See Fig. 16). The concentrated masses at 
each panel point are tabulated in Table 5. 











lO 












sO 






rH 




i-H 


lO 






in 


O 




■H 


















o 


o 


o 








SO 


















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• — 1 




i-H 


n 








o 


iD 


■— t 


















o 


o 


s0 












CO 














*f 


















o 




o 








co 


o 


co 




















CO 












vO 


















in 












o 






<— 1 




rH 


in 






in 


o 




• — i 




CO 








• 






o" 


o 


d 








SO 




































r- 




!> 


\Q 








o 




rH 




CM 


» 


• 


• 








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* 




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o 


o 


o 


CO 




>H 


• 












o 




00 












CM 




u 








p-H 




o 






r-l 






4-> 






ro 






ro 






In-— • 


-t-> 




i — i 


4-> 




3 • 


_C 




<H 


C 


jC • 


+j xi 


cjn 


1 


•H 


•H 




O H 


•H 




U 


O 


■H i— ( 




0) 


J3 • 


10 


Du 




H 


3B 


i-H C 


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+-> +» 




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in 


w JS 




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94 



Table 4. ANALYTICAL AND EXPERIMENTAL 

NATURAL FREQUENCIES IN RAD ./SEC 
(TEST NO. 1) 





Mode 


Analytical 


Experimental 


Error % 


1 


85.43 


90.8 


- 6.3 


2 


387.05 






3 


785.61 






4 


1111 .0 






5 


1378.0 






6 


1534.10 







+J 
c 

■H 
O 
Oh 



03 

c 

ro 

pu 



rH 


o 


rH 


if) 


o 


if) 








d 


CO 


CO 


sO 


o 






rH 


rH 


i— i 


* 

o 


, 1 


in 


o 


if) 


d 


, — 1 


, j 




1 — 1 






1 — 1 


rH 








o 


o 


O 


CO 


o 


CO 


co 


CM 


if) 


■■o 


o 






rH 


rH 




* 




<— 4 


o 


H 


If) 


o 


If) 


0 

O 


EN 


CN 




o 


vO 






r-( 




* 






o 


h- 


if) 


o 










00 


d 


d 


sD 




r- 




rH 






* 




O 




o 



o 



00 
CM 



m 

CM 

"XT 



00 

CI 



o 

CO 
CM 

vj- 



in 
o 

CM 



rH 
It 



o 

vO 

CM 
CO 























M 


JZ 










O 


O) 




I XI 






4-> 


•H 




U r-t 






ro 


CD 


+->-—- 


3 — • 


x: 






5 




+■> 












O 4-> 


■H 




•H 


"0 


•H r-( 


3 x: 


CD 


X) C 


O 


ro 


<u — - 


r-l a> 




rH -H 


05 


Q) 




4-> -H 




— X 


O 


rj 


w 


co a> 




CM 




T5 l/) 


• 3E 


ro • 


0) . 


* 


* 


fD 3 


C 


X} 


U) o 


* 


<U M 


O rH 


O rH 








Q N 


S3 ro 




S w 







96 



The analytical and experimental natural frequencies 
of the test are compared in Table 6. 



Table 6. ANALYTICAL AND EXPERIMENTAL 

NATURAL FREQUENCIES IN RAD. /SEC. 
(TEST NO. 2) 



Mode 


Analytical 


Experimental 


Error % 


1 


66.40 


65.70 


+ 1.05 


2 


263.06 


(236.93)* 


( +10.00)* 


3 


513.58 






4 


707.34 






5 


846.97 






6 


936.26 







* These values will not be considered as experimental 
results as the error is too large. 



During the experiment, it was found that the clamps 
which provided rigid connection between the lead weights 
and the panel joints did not function properly at higher 
frequencies and produced impact on the test tower. Thus, 
the 10 per cent error found in this test is due in part 
to these side affects. 

In the third test, three pieces of 20 lb. rail 
each 1 ft. -9 in. long were bolted to the web members of 
the test tower at panel points 2 through 5. Two pieces of 
rail were bolted at the lower part of the web member at 



97 

panel point 6 (See Fig. 17). The concentrated masses of 
this system are tabulated in Table 7. 

Table 7. DISTRIBUTION OF MASSES 
(TEST NO. 3) 



Panel Point 



Dead 



Weight 60.0 63.57 60.51 62.39 60.51 60.51 

Truss 

(lb.) 

Non- struc- 
tural weight ** ^ # 
( lb -) 64.0 108.70 107.50 108.50*107.60 77.50 

Total 

Weight 124.0 172.27 168.01 170.89 108.11 138.01 

{lb, ) 



Mass (lb, - 

sec. 2 /in. .3260 .4717 .4606 .4681 .4609 



,3584 



** Oscillator 
* Rails 



During the third test the non- structural masses appeared to 
move in harmony with the parts of the structure to which 
they were attached. The analytical and experimental 
natural frequencies are tabulated in Table 8. 



98 



Table 8. ANALYTICAL AND EXPERIMENTAL 

NATURAL FREQUENCIES IN RAD. /SEC. 
(TEST NO. 3. ) 



Mode Analytical c Experimental Error % 

1 64.84 65.00 -1.02 

2 259.05 267.00 -2.93 

3 513.38 

4 707.12 

5 847.47 

6 .916.56 



6.9. Discussions of the Test Results 

a) The errors in the natural frequencies found in the 
experiments ranged from one to 6 per cent in the 
fundamental mode and 3 per cent in the second 
mode. These figures are considered to be small 

in practice, and they also indicate that the 
mathematical model formulated in Chapter II is 
adequate to represent the real structure and yield 
accurate results on lower modes when suppport 
flexibilities are known. 

b) The Stodola-Vianello procedure appears to be the 
simplest method available to obtain the desired 
natural frequencies for truss structures. 



99 



The accuracy of the computed lower modes is 
quite satisfactory. 

c) The non-structural masses that are a part of the 
real or test structure should be attached rigidly 
to the structural members so as to avoid possible 
induced impact at high vibration frequencies. 

d) The fact that the natural frequencies of the real 
structure are higher than those of the mathematical 
model can be attributed in part to the affect of 
rigid construction at the joints which tends to 
stiffen the trusses. Also, rotary inertia effects 
may have influenced the results. 

e) The analytical analysis on natural frequencies of 
the mathematical model used in the Test No. 1 in- 
dicated that the flexibility of support has signi- 
ficant influence over the first and the second 
natural frequencies and very little on frequencies 
higher than third mode. The relationship between 
the support flexibility and the natural frequencies 
of the mathematical model is shown in Fig. 18. 



where 



100 



"V _ ^11 ) support 
^11 ) "truss 



(d^^) support s deflection at panel point 1 due to a 

unit load at panel point 1 considering 
the flexibility of the support 
= displacement of panel point 1 due to a 
unit load applied statically at panel 
point 1, 



(d-Q) truss 



and 



f 



( ^ n )f lexible base 
n )fixed base 



101 




. SIX-MASS SYSTEM (TEST NO. 1) 

EIGHTEEN-MASS SYSTEM (CHAPTER VII) 



CHAPTER VII 



NUMERICAL ANALYSIS OF STEEL TOWER SUBJECT TO WIND 

The proposed method developed in Chapter V has 
been applied to three structural systems, namely a 
three-mass system, a six-mass system and an eighteen-mass 
system. The IBM 709 Computer was used for computing 
the normal modes, their associated natural frequencies, 
and the displacement responses. The computation proce- 
dures used are described as follows: 

1) A computer program for the Stodola-Vianello method 
was used to obtain the normal modes and their 
associated natural frequencies. (see Appendix B) 

2) Random wind velocities similar to those presented 
in Sherlock's wind map were used to generate the 
wind pressure at each mass point. The computer 
program listed in Appendix A was used to obtain 
the displacement responses. 

3) The output cards obtained from the previous calcu- 
lation were used to plot the response curves by a 
Calcomp 563 Plotter available at the University of 
Florida Computing Center. 

Notations used throughout this Chapter may be expressed 
in matrix forms as follows: 

102 



103 



mass at panel point i 



= flexibility factor as defined in Chapter III 



i*" natural frequency, i.e., i is the funda- 
mental frequency 



k^j = wind conversion factor at mass i as defined 
in Chapter I 
^$^ m = normal ordinates of mass i at mode m 



where 



1 represents the column matrix, and [ "] the nxn 



square matrix. 



7.1 The Three- Mass System 



Win 

m 3 



/7T77r 



Fig. 19 THREE- MASS SYSTEM 

The mathematical model as shown in fag. 19 has 
the following numerical values: 
The conversion factor matrix is 

1 



1 
1 



104 



The mass matrix is 



416 1 
mAx.1 832 



I 1 i [ 832 



i lb. -sec. 2 /in. 



The flexibility matrix is 

. [22.70 15.18 7.81 

dj A* 15.18 14.33 7.74 

L JJ 7.81 7.74 6.94 



x 10" 6 in./lb. 



By the Stodola-Vianello method, the normal modes and 
their associated natural frequencies were obtained as, 
f ol 1 ows : 



^ im ] - 



0.02874 -0.03050 0.02540 
0.02412 0.00210 -0.02467 
0.01424 0.02704 0.01633 



(in. /lb. -sec. 2 )" 2 



and 



1 28.73 



rad./ sec 



J 



The wind velocities at each mass point for the three- 
mass system are listed in Table 9. Based on the wind 
velocity of Table 9, the displacement responses of 
masses 1, 2, and 3 computed by the proposed method are 
plotted in Fig. 20 in which all three modes are used. 
The displacement curves indicate that the complete 



105 

system is under a motion of fundamental mode. In order 
to examine the effects of different combination of modes* 
on the displacement curves, the displacement of indi- 
vidual masses 1, 2, and 3 are plotted in Fig. 21, 22, and 
23 respectively. It is apparent that for this specific 
system, the fundamental modes yield a quite satisfactory 
result. 



The term modes used throughout this Chapter 
will be referred to as the normal modes and their 
corresponding natural frequencies. 



106 



Table 9 

WIND VELOCITIES AT EACH MASS POINT 
(THREE-MASS SYSTEM) 



Mass 1 Mass 2 Mass 3 



Time 
(sec.) 


Velocity 
(m.p.h) 


Time 

(sec.) 
— 


Velocity 
(m.p.h) 


Time 
(sec . ) 


Veloc ity 
(m. d . h . ) 


0.0 


79.0 


0.0 


71.0 


0.0 


63.0 


0.4 


78.0 


0.4 


70.0 


0.5 


61.0 


0.7 


75.0 


0.8 


81.0 


0.8 


51.0 


0.8 


62.0 


1.3 


79.0 


1.0 


74 0 


1.5 


92.0 


1.6 


60.0 


1.3 


62.0 


1.7 


95.0 


2.0 


86.0 


1.6 


78.0 


2.0 


80.0 


2.3 


64.0 


; 2.0 


75.0 


2.4 


83.0 


2.8 


57.0 


2.6 


62.0 


2.8 


70.0 


2.9 


83.0 


3.0 


60.0 


3.0 


95.0 


3.4 


79.0 


3.6 


75.0 


3.4 


98.0 


3.5 


56.0 


4.0 


74.0 


3.7 


90.0 


3.9 


78.0 


4.3 


65.0 


3.9 


70.0 


4.3 


83.0 


4.9 


87.0 


4.0 


83.0 


4.8 


77.0 


5.3 


89.0 


4.4 


81.0 


5.0 


88.0 


5.7 


60.0 


4.6 


91.0 


5.5 


61.0 


6.0 


50.0 


5.0 


50.0 


5.9 


90.0 


6.5 


55.0 


5.5 


86.0 


6.3 


85.0 


7.0 


66.0 


5.8 


80.0 


6.8 


70.0 


7.6 


67.0 


6.4 


85.0 


7.3 


101.0 


8.0 


70.0 


6.9 


77.0 


7.7 


91.0 


8.5 


65.0 


7.2 


83.0 


7.9 


69.0 


8.9 


68.0 


7.5 


84.0 


8.3 


77.0 


9.3 


50.0 


7.8 


97.0 


8.7 


73.0 


9.7 


91.0 


7.9 


70.0 


9.3 


88.0 


10.2 


80.0 


8.3 


97.0 


10.2 


60.0 






8.5 


90.0 










8.8 


91.0 










9.0 


70.0 










9.4 


95.0 










9.7 


93.0 










10.0 


70.0 











110 




* 



7.2. The Six-Mass Svst.pm 



m. 



111 



m 3 



m 4 

m. 



Fig. 24. SIX-MASS SYSTEM 

The mathematical model as shown in Fig. 24 has 
the following numerical values: 
The conversion factor matrix is 



i = 



1 
1 
I 
1 

U J 



The mass matrix is 



■ 



( 0.3667^ 
0.6734 

J 0.6734 

\ 0.1554 
0.1554 

v 0.1554 



} (lb.-sec. 2 /in.) 



112 



The flexibility matrix is 



265.15 
206.77 
150.82 
99.72 
55.91 
21.81 



206.77 
172.00 
127.86 
86.14 
49.29 
19.72 



150.82 
127.86 
104.90 
72.57 
42.67 
17.62 



99.72 
86.14 
72.57 
58.99 
36.04 
15.52 



55.91 
49.29 
42.67 
36.04 
29.42 
13.43 



21.81 
19.72 
17.62 
15.52 
13.43 
11.33 



xl0- 3 in./lb« 



By the Stodola-Vianello method, the normal modes and their 
associated natural frequencies were obtained as follows: 



59.40 ^ 
316.90 
621.09 
715.64 
1132.74 
^1482.50 J 



rad. /sec . 



and 



J* 
im 



0.94592 0.97351 0.74725 0.57109 0.00104 0.00359 
0.77447 0.10097 -0.51735-0.77795 -0.04534 0.00935 
0.58875 -0.74844 -0.26541 0.64811 0.27616 -0.10702 
0.40033 -0.93269 0.81996 -0.15115 -1.73140 1.30850 
0.23110 -0.80900 1.13944 -0.81703 -0.11027-1.76150 
0.09330 -0.45023 1.01760 -0.72959 1.17567 1.25300 



(in. /lb. -sec. 2 )" 2 



The wind velocities at each mass point for the six-mass 
system are listed in Table 10. 



113 



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114 

From the result of the computation of Table 10, it is 
found that for masses 1 through 5, the displacement 
response shows no significant change from use of the 
fundamental mode to use of all the modes. But for mass 
6, at least the lowest three modes are required to 
achieve the same result. The response curves for the 
entire structure are shown in Fig. 25. The response 
curves of mass 6 using the fundamental mode, the lowest 
two modes, and all three modes are plotted, respectively, 
in Fig. 26. 



116 




(*UT £-01) ^uatuaoexdsfa 



7 • 3 The Eighteen- Mass System 

An actual steel tower having eighteen panels was 
studied. A sketch of that tower and its equivalent 
mathematical model are shown in Fig. 27. 




Fig. 27. THE EIGHTEEN- MASS SYSTEM 



118 



The numerical values of the eighteen-mass system are as 
follows: The conversion factor matrix is 



k i= 



( 1680.60 \ 
871.40 
871.40 
871.40 
871.40 
871.40 
871.40 
871 .40 
871.40 
871.40 > 
871.40 
871.40 
871.40 
871.40 
871.40 
894.20 
1249.00 

U697.20 



I 



The mass matrix is 



m. 



344 
168 
150 
159 
153 
160 
189 
175 
205 
235 
217 
223 
260 
263 
336 
378 
428 



.35 > 
.78 
.98 
.32 
.96 
.51 
.33 
.40 
.47 
.90 
.51 
.84 
.44 
.58 
.57 
.42 
,48 



(lb. -sec , 2 /in. ) 



119 



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121 





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in 




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rv 


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r-i 


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pn 




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f\i 


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en 










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Lf>. 


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in 




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oooooooooooooooooo 



122 

Note the peculiar characteristic of this flexi- 
bility matrix in which some of the elements are negative. 
This is caused by the splayed configuration near the base 
and the exceptionally heavy chord members that occur 
there. The symbol E appearing in the numerical values 
represents the exponetial sign. 

Due to round-off errors and difficulty with con- 
vergence, the highest mode obtained by the Stodola- 
Vianello was mode 11 even when double precision arithmetic 
was used in the computer. Hence, all the subsequent com- 
putations will be limited to a maximum of eleven modes. 
The natural frequency matrix is: 



4.63 
11.19 
19.39 
30.77 
37.19 
44.81 
51.72 
58.83 
66.33 
72.50 
78.05 

not found 



> 



M 

II 

If 



rad./ sec 



[^not found 

The normal modes (modes 1 through 11) of the eighteen-mass 
system are given in the following pages. The symbol E as 
appearing in the numerical values represents the exponen- 
tial sign. The wind velocities are shown in Table 11. 



oooooooooooooooooo 

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Jjj -'O-' , M- < >\-<-''-i-0l/>>-'i-(-i(flG0f\ii-l 



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127 

The displacement responses of masses 1 through 18 
axe listed in Table 12 through Table 29. The combination 
of the lowest mode through the fifth mode and through the 
tenth mode are illustrated. Although the entire eighteen 
modes were not included, the results from the ten-mode 
computation are considered to be essentially an exact 
solution. This conclusion follows from a comparison of 
the results obtained from the five-mode through the ten- 
mode computations. The data obtained from using the lower 
modes indicate that in order to achieve the same accuracy 
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1) For masses 1 through 9, the desired accuracy could 
be obtained by using only the fundamental mode. 

2) For masses 10 through 17, the lowest two modes are 
required. 

3) Fox mass 18, the lowest five modes are required. 
For towers that vary more uniformly along their height, 
acceptable results might be obtained for all masses using 
only the first two modes. 

The response curves of the entire structure using 
ten modes are plotted in Fig. 28. Response curves for 
masses 1 through 4 are plotted in Fig. 29 through Fig. 32. 
mass 9 in Fig. 33. masses 15 through 18 in Fig. 34 through 
Fig. 37. respectively, in which different combinations of 
modes are illustrated. 



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CHAPTER VIII 



SUMMARY, CCNCLUSICNS AND RECOMMENDATIONS 
FOR FUTURE RESEARCH 

8.1. Summary 

The displacement responses of truss-type tower 
structures, cantiievered from their bases and free from 
any intermediate supports, were studied. The flexibility 
coefficients at each panel point of a tower were obtained 
by methods used in the conventional truss analysis, the 
following assumptions being used. 

a) The members are connected by fric tionless pins. 

b) The length L of each individual bar is taken as 
the distance between the centers of its end 
joints . 

c) The cross-section of each bar is taken as con- 
stant over the length L. 

d) The elongation of each bar is given by SL/AE. 

e) The loads are applied only at the joints. 

f) The system is linear. 

In order to study the dynamic characteristics of 
a real structure, a mathematical model based on the 
following assumptions was formulated. 



174 



175 

a) The mathematical model is a concentrated-mass 
system which possesses the identical linear 
elastic properties of the real structure. 

b) The damping is not considered. 

c) The law of superposition holds. 

d) The masses are lumped at each panel point. 

e) Rotary inertia effects of individual members 
and equipment are neglected. 

f) The dynamic forces which are piecewise continuous 
functions acting on each panel point can be 
represented either analytically or numerically. 

The numerical solutions for the displacement response of 
the mathematical model can be expressed in terms of the 
normal modes, their associated natural frequencies and 
the derivatives of the forcing functions. Numerical 
solutions using the adopted method were found for a 
three-mass system, a six-mass system, and an eighteen- 
mass system, each subjected to a formalized wind function. 
An IBM 709 Computer was used to compute the displacement 
responses of each system, and a Calcom 536 Plotter was 
used for plotting all the response curves. 

To verify the dynamic similarity between a real 
structure and its mathematical model, steady-state experi- 
ments were conducted on a steel tower of 15 foot length 
with k-type diagonal bracing and a rectangular cross 



176 

section of 2 ft. 0 in. by 1 ft. 4 in. The test results 
showed that the computed natural frequencies of the 
mathematical model for the first and second modes checked 
closely with those obtained from the experiments. Since 
the maximum range of the alternating force generator used 
is limited to 3600 rpm, natural frequencies of the test 
tower from the third mode up could not be obtained. 

8.2. Conclusions 

a) A real truss-type structure with non- structural 
masses rigidly attached at the intersections of 
its structural members can be represented by a 
mathematical model which is a concentrated-mass 
system, and the natural frequencies of that 
system can be obtained by the Stodola-Vianello 
procedure . 

b) The results of Test No. 3 showed that for a six- 
mass system there was a 1 per cent error in the 
calculated fundamental frequency and a 3 per 
cent error in the second mode frequency. These 
figures seem to justify the statement that the 
lower natural frequencies obtained from an 
appropriate mathematical model will be reasonably 
accurate. Although the experiment did not carry 
beyond the second mode, we may presume on the 
conservative side that the lowest one- third of the 



177 

lower natural frequencies and their associated 
normal modes obtained from the mathematical model 
are sufficient for use in the numerical method of 
calculating dynamic responses. 
From the results of the numerical calculations, 
it can be said that not all the natural frequencies 
and their associated normal modes of a mathematical 
model are required for computing the dynamic 
response. In the case of three- and six-mass 
systems, the use of the fundamental frequency and 
its associated normal mode gave excellent accuracy 
at each mass point. For the eighteen-mass system, 
the use of fundamental mode yielded an accurate 
response only for masses 1 to 9- in the upper 
half of the tower: when the lowest two modes were 
used, the accuracy extended downward to mass 17; 
and when the lowest three modes were used, the 
response of mass 18 at the first panel point 
above the base of the tower could be determined 
with sufficient accuracy. However, the use of 
only the fundamental mode in predicting the 
dynamic response for a structural tower can result 
in serious errors in the displacements of the 
lower half of the structure particularly if the 
tower changes section rapidly at one or more 
panels . 



178 

The xound-off error in applying the Stodola- 
Vianello procedure to the eighteen-mass system 
was considered to be significant. In the process 
of computing the natural frequencies by that 
system, the convergent criterion for the assumed 
mode and the computed mode was set at 10" and 
the double precision procedure was used in the 
computer program. It was found that one signifi- 
cant figure was lost for each frequency computed; 
therefore, the highest natural frequency obtained 
in this manner was limited to the eleventh mode. 
The computer time required to obtain the eleventh 
mode was recorded to 0.5 hour whereas only 0.1 
hour was required for computing the first three 
modes. 

The displacement response of the first concen- 
trated mass above the cantilever base, i.e., 
mass 6 in the six-mass system or mass 18 in the 
eighteen-mass system, was of small magnitude. 
Any movement of the other masses in the struc- 
tural system would have a significant influence 
on its displacement. Therefore, in order to 
achieve the same degree of accuracy for the 
response of mass six as for the other masses, 
the use of more modes is required in the compu- 
tation. 



179 



The computer time required for computing the 
dynamic response of an eighteen-mass system for 
an eight-second period was estimated as follows: 



The degree of accuracy of the displacement 
responses based on a five-mode computation was 
judged to be as good as those obtained from the 
ten-mode computation. Hit the time spent on the 
ten-mode computation was twice as much as for 
five modes. Hence, it is important to decide 
in advance as to the number of modes required for 
acceptable accuracy. 

The numerical procedure applied in this study 
allows the degree of accuracy to be controlled by 
the dynamic characteristics of the mathematical 
model and the number of fundamental modes employed 
rather than by the time interval which is the 
critical factor in most of the trial- and-error 
procedures. Thus, the problem of deciding a 
suitable time interval to satisfy the convergence 
and stability criteria is largely avoided. 



Number of modes use 



Time required (hr) 



1 
2 
3 
4 
5 
10 



.04 
.08 
.12 
.16 
.20 
.40 



180 

8.3. Recommendations for Future Research 

Although a satisfactory numerical procedure has 
been derived in this study to predict the dynamic response 
of a lumped-mass mathematical model to random disturbances, 
there are still many important problems that must be 
solved before the probable safety of real tower structures 
can be computed. Some of the more important areas that 
need study are as follows: 

a) The results of this limited study indicate that 
the fundamental mode of vibration is the pre- 
dominant one for a practical structure subjected 
to varying wind velocities similar to those 
recorded by anemometers. Nevertheless, several 
higher modes may be required for study of the 
response of the lower panel points of very tall 
towers or for unusual shaped towers. Therefore, 
it is important to have the capability of calcu- 
lating the higher modes of the real structure 
with accuracy. Procedures need to be developed 
that will: 

1) Take into account the inertia effects caused 
by displacements of the truss joints along 
all three coordinate axes. 

2) Take into account the mass moment of inertia 
of each truss member. 



181 

3) Take into account the mass moment of inertia 
of each piece of attached equipment. 

4) Allow the computation of the higher modes 
without the difficulties of round-off errors 
and convergence that are experienced when 
the Stodola-Vianello procedure is used. 

The influence of longitudinal and transverse 
vibrations of the individual members should be 
studied to discover the circumstances under which 
these effects take on major importance. 
The most important and also the most difficult 
problems that must be solved are those concerned 
with the true character of natural extreme winds 
and the aerodynamic properties of the real struc- 
ture. Some of the most pressing problems are: 

1) What is the correct form of the variation of 
wind velocity with height? This characteris- 
tic appears to vary with geographic location 
and mean wind velocity. 

2) What are the spatial dimensions of the micro- 
gusts that have been recorded by anemometers? 
This factor will determine the extent to 
which a tower is engulfed by a gust. If the 
spatial dimensions of the micro-gusts are 
small, then only short sections of the various 



182 

truss members in a tower may be affected by 
each gust. Due to the random nature of the 
gusts, the integrated effects of many small 
gusts over a reasonably large truss panel 
might be a relatively steady force with corre- 
sponding reduction of the dynamic responses 
predicted in this dissertation. 
How long does it take for dynamic pressures 
to readjust to new wind velocities when those 
velocities change as in a micro-gust? The 
time required for readjustment probably 
depends on the dimensions of the wind area 
affected and the relationship of those 
dimensions. If the time lag involved is 
relatively long compared to the time between 
gusts, the full effects of gusts will not 
develop and the wind force will not vary with 
time in the same manner as the square of the 
wind velocity. The response of the structure 
is a function of the wind force magnitude and 
its distribution. 

What is the proper procedure for computing 
the wind force on an open trussed tower? 
Individual shape factors, mutual shielding, 
and gross changes in the wind pattern near 



porous obstructions are effects that render 
the exact calculation of wind forces impos- 
sible under the present state of the art. 
The answer to these questions will be difficult and 
expensive to obtain. Even the reasonably great effort 
being made by the Bookhaven National Laboratory has not 
provided the data required. The answers to Questions 3 
and 4 may possibly be found from properly conducted wind 
tunnel tests . 



184 



BIBLIOGRAPHY 



1. Bickley, W. G. "Numerical Differentiation Formulae 

for Numerical Differentiation," Mathematical 
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Report of the Task Committee on Wind Forces of 
the Committee on Loads and Stresses of the 
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3. Boley, B. A. and Chao, C. C. "Impact on Pinjointed 

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Jan. , 1955. 

4. Bolotin, V. V. The Dynamic Stability of Elastic 

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5. Buckingham, R. A. Numerical Methods. London: Pitman, 

1962, p. 224. 

6. Cohen, E. "Discussion of Variation of Wind Velocity 

and Gusts with Height," Proc. ASCE , Vol. 79, 
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7. Chiu, A. N. L., Sawyer, D. A. and Grinter, L. E. 

"Vibration of Towers as Related to Wind Pulses, 
Journal. ASCE , Vol. 90, No. ST5, Oct. 1964, 
pp. 137-160. 

8 Collins, G. F. "Determining Basic Wind Loads," 

Proc, ASCE, Vol. 81, Paper No. 825, Nov. 1955. 

9. Contractor, G. P. and Tompson, F. C. "The Damping 
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10. Davenport, A. G. "Application of Statistical 

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11. Davenport, A. G. "The Response of Slender, Line-Like 

Structures to a Gusty Wind," Proc. . Inst, o f 
Civil Eng. . Vol. 23, Nov., 1962, pp. 389-408. 



185 

12. Duncan, W. J. "A Critical Examination of the Repre- 

sentation of Massive and Elastic Bodies by 
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13. Ellington, J. P. "The Vibration of Segmented Beams," 

Brit. Journal of Applied Physics , Vol. 7, No. 8, 
Aug. 1956. 

14. Foppl, 0. "Practical Importance of Damping Capacity 

of Metals, Especially Steels," Iron and Steel 
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15. Livesley, R. K. "The Equivalence of Continuous and 

Discrete Mass Distributions in Certain Vibration 
Problems," Quarterly Journal of Mech. and Appl. 
Math. , Vol. 5, Pt. 3, 1953, pp. 353-360. 

16. Milne, W. E. Numerical Calculus ,, Princeton: Princeton 

University Press, 1949, pp. 131-144. 

17. Morris, J. and Head, J. W. "The Escalator Process 

for Solution of Lagrange's Frequency Equations," 
Phil. Mag. . Vol. 35, No. 350, Nov. 1944, p. 735. 

18. Myklestad, N. 0. "A New Method of Calculating 

Natural Modes of Uncoupled Bending Vibration of 
Airplane Wing and Other Types of Beams," Journal 
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19. Neubert, V. H. "Lumping of Mass in Calculating 

Vibration Responses," Proceedings. ASCE , 
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20. Newmark, N. M. and Chang, S. P. "A Comparison of 

Numerical Methods for Analyzing the Dynamic 
Response of Structures," Univ. Illinois Civil 
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21. Newmark, N. M. and Tung, T. P. "A Review of Numerical 

Integration Methods for Dynamic Responses of 
Structures," Univ. Illinois Civil Eng. Studies , 
Structural Research No. 69 , 1954. 

22. Newmark, N. M. "A Method for Computing for Structural 

Dynamics," Trans. , ASCE , Vol. 1, 1962, 
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186 



23. Nielsen, N. N. "Steady-State Versus Run-Down Tests 

of Structures," Proceedings, ASCE, Vol. 90, No. 
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24. Norris, C. H. et al . Structural Design for Dyna mic 

Loads , New York: McGraw-Hill, 1959, pp. 73-212. 

25. Saffir, H. S. "The Effects on Structures of Winds 

of Hurricane Force," Proceedings. ASCE , Vol. 79, 
Sep. No. 206, July 1953. 

26. Scarborough, J. A. Numerical Mathematical Analysis , 

2d ed. Baltimore: John Hopkins Press, 1950. 

27. Scruton, C. S. and Newberry, C. W. "On the Estima- 

tion of Wind Loads for Building and Structural 
Design," Journal. Inst, of Civil Eng. , Paper 
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28. Sherlock, R. H. "Variation of Wind Velocity and 

Gusts with n eight," Proceedings. ASCE . Vol. 78, 
Sep. No. 126, April 1952. 

29. Singer, I. A. and G. S. Raynor. "Analysis of 

Meteorological Tower Data, April 1950-March 1952," 
Brookhaven National Laboratory, New York, June, 
1959. 

30. Singer, I. A. and G. S. Raynor. "A Study of the 
Wind Profile in the Lowest 400 Feet of the 
Atmosphere, Progress Report No. 3," Brookhaven 
National Laboratory, New York, June 1959. 

31. Singer, I. A. and G. S. Raynor. "A Study of the Wind 
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Progress Report No. 4," Brookhaven National 
Laboratory, New York, Oct. 1959. 

32. Singer, I. A. "A Study of the Wind Profile in the 

Lowest 400 Feet of Atmosphere, Progress Report 
No. 5," Brookhaven National Laboratory, New 
York, June 1960. 

33. Singer, I. A. and Leo, J. Tick. " A Study of the 

Wind Profile in the Lowest 400 Feet of Atmosphere, 
Progress Report No. 6," Brookhaven National 
Laboratory, New York, May 1960. 



187 



34. Singer, I. A. and C. M. Nagle. "A Study of the Wind 

Profile in the Lowest 400 Feet of Atmosphere, 
Progress Report No. 7," Brookhaven National 
Laboratory, Sept. 1960. 

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Proceedings. ASCE , Vol. 80, Sep. No. 539, 
Nov. 1954. 

36. Timoshenko, S. and Young, D. H. , Vibration Problems 

in. Engineering , New York: McGraw-Hill, 1956, 
pp. 143-145. 

37. Van Erp, J. W. T. "Wind-Load Standards in Europe," 

Proceedings. ASCE . Vol. 76, Sep. No. 42, 
Nov. 1950. 

38. Weaver, W. Jr. and Kane, T. R. "Dynamics of a Large 

Steerable Radio Telescope Antenna," Proceedings . 
ASCE . No. ST4, Aug. 1963, pp. 55-69. 



J 



APPENDIX A 



COMPUTER PROGRAM FOR COMPUTING DYNAMIC 
RESPONSE OF STEEL TOWERS 



189 



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BIOGRAPHICAL SKETCH 

George Chien Kao was born on June 22, 1933, in 
Shanghai, China. He graduated from the Provincial 
Kao-hsiung Middle School, Kao-Hsiung, Taiwan in 1952. 
In October, 1952, he entered the National Taiwan Uni- 
versity and was graduated in June, 1956 with B. S. 
degree in Civil Engineering. He then received Reserve 
Officer's training in the Air force for a period of 
eighteen months. In March, 1958, he entered the Uni- 
versity of Minnesota as a graduate student in Structural 
Engineering and was graduated in June, 1960 with the 
degree of M. S. in Civil Engineering. He then worked 
for the Osboro Engineering Company, Cleveland, Ohio, as 
a structural designer. As of January, 1962, he was 
granted a leave of absence for advanced work toward 
the Ph. D. degree at the University of Florida. 

He is married to the former Bern ice Tze-wei Chow. 
He is an Associate Mamber of the American Society of 
Civil Engineers, and a registered civil engineer of the 
Republic of China. 



This dissertation was prepared under the direction 
of the chairman of the candidate's supervisory committee 
and has been approved by all members of that committee. , m 
It was submitted to the Dean of the College of Engineering 
and to the Graduate Council and was approved as a partial 



fulfillment of the requirements for the degree of Doctor 
of Philosophy. 

April 24 , 1965 




Dean, College of Engineering 



Dean, Graduate School 



SUPERVISORY COMMITTEE 





Co- Chairman